ABSTRACT An important contribution made by Canadian mathematicians to Lie theory in the last decade and a half involves their work on extended affine Lie algebras, which were first introduced by the physicists Høegh-Krohn and Torrésani in their work on quantum gauge theory.
We examine the finite-dimensional irreducible representations of the Lie algebra 2n(Cq[t11t21]), where =1 and Cq[t11t21] is the associative algebra of non-commuting Laurent polynomials over C with t2t1=qt1t2 with q=1. (When =-1 and n2, this Lie algebra is an example of an extended affine Lie algebra of type Cn.)
This is joint work with Hongjia Chen and Professor Yun Gao.
ABSTRACT: A permutation p of 12N is said to contain a pattern (relatively shorter permutation) q of length k (kN) if p contains a substring of length k that has the same relative order as q. Let SN(q) denote the set of permutations of length N which avoid the pattern q. In this talk, I will present a brief sketch of the proof of a ratio limit theorem for the number of q-avoiding permutations when q belongs to some specific classes. Considering a permutaion of length N as a set of N points in the xy-plane, I will also discuss some results about the typical shape of some q-avoiding permutations in the xy-plane.
This is joint work with my postdoc supervisor Neal Madras.
ABSTRACT: I'll define drift configurations and use them to give compatible and new combinatorial rules for Kazhdan-Lusztig polynomials and (Hilbert-Samuel) multiplicities of vexillary Schubert varieties.
This is joint work with Li Li (U. Illinois at Urbana-Champaign).
ABSTRACT: I will review the combinatorics of increasing tableaux, as introduced by Alex Yong and me, and recall how this combinatorics can be used to give a formula for the structure constants for the K-theory of Grassmannians. I will then explain how shifted increasing tableaux can be used similarly to provide a formula for the structure constants of the K-theory of odd orthogonal Grassmannians. An essential ingredient in our proof is the Pieri rule for K-theory of odd orthogonal Grassmannians, recently obtained by Buch and Ravikumar. The other ingredients are essentially tableau combinatorics; they will be the focus of my talk. The work to be described is all joint with Alex Yong, and is contained in the preprints arXiv:1002.1664 (for odd orthogonal Grassmannians) and arXiv:0705.2915 (for usual Grassmannians).
ABSTRACT: I will give a combinatorial formula for certain coefficients of the operator when it acts on a Hall-Littlewood symmetric functions. This result (almost completely) answers a conjecture posed by Alain Lascoux in the paper by (F)Bergeron-Garsia-Haiman-Tesler that introduced the operator . The combinatorial formula is proven by showing that q,t-counting Dyck paths satisfy the same recursion as a symmetric function expression. This is joint work with N. Bergeron, F. Descouens, A. Garsia, J. Haglund, A. Hicks, J. Morse and G. Xin.
ABSTRACT: Misra and Miwa gave a realization of the crystal of the basic representation in which the nodes of the graph are indexed by "p-regular" partitions. These "p-regular" partitions also index irreducible representations of the symmetric group over a field of characteristic p. After reviewing this, I will talk about some other partition based models for this crystal and their connections to the modular representation theory of the symmetric group.
ABSTRACT: Indecomposable permutations (also called connected) are considered in many textbooks in combinatorics. Their number in $\S_n$ was probably found for the first time by Marshall Hall around 1950; it was then proved that it is close to $n! - 2/n$ for $n$ big, showing that almost all permutations are indecomposable.
In this talk we will show that this fact depends heavily on the number of cycles of the permutation. A connection with the enumeration on maps in oriented surfaces will also be evoked.
ABSTRACT: Conformal superalgebras describe symmetries of superconformal field theories and come equipped with an infinite family of products. They also arise as singular parts of the vertex operator superalgebras associated with some well-known Lie structures (e.g. affine, Virasoro, Neveu-Schwarz).
In joint work with Arturo Pianzola and Victor Kac, we classify forms of conformal superalgebras using a non-abelian Cech-like cohomology set. As the products in scalar extensions are not given by linear extensions of the products in the base ring, the usual descent formalism cannot be applied blindly. As a corollary, we obtain a rigourous proof of the pairwise non-isomorphism of an infinite family of N=4 conformal superalgebras appearing in mathematical physics.
ABSTRACT: We will use the twisted Heisenberg Lie algebras to construct a family of vertex operators. These will provide representations for certain Steinberg unitary Lie algebras coordinatized by quantum tori. As a by-product we recover some of affine Kac-Moody Lie algebras.
This is a joint work with N.Jing and S.Tan.
ABSTRACT: I will show that the pre-Lie operad is a free non-symmetric operad. I will do some recall on the notions of Operads.
This is joint work with M. Livernet.
ABSTRACT: Let G be a simple complex finite-dimensional Lie algebra, and L=L(G) a Loop algebra corresponding to a diagram automorphism of G. These algebras occur as the main in a construction of the Affine Lie algebras, and their representation theory is subject of continuing interest. Some aspects of the finite-dimensional representation theory of the 'twisted' loop algebras are now well understood. In particular, the universal 'loop-highest weight' Weyl modules have recently been described, has have the blocks of the corresponding (non-semisimple) category.
We will discuss these recent classifications, as provide a more geometric reformulation of the results. We will then discuss possible extensions of this theory to the multiloop generalizations of L, and beyond.
ABSTRACT: In 1990, the LLT polynomials were defined as q-analogs of products of Schur functions. In this talk, we define new analogs of LLT polynomials on the non-commutative side. We give an analog of the quotient map, and we also introduce an interpretation of these new polynomials in terms of representations of the Hecke algebra. This non-commutative approach could give us some ideas for solving some problems on the commutative side.
ABSTRACT: Let V be a vector space over the complex with basis {x1,x2,...,xn} and G be a finite subgroup of GL(V). Then the tensor algebra T(V) of V over the complex is isomorphic to the polynomials in the non- commutative variables x1, x2, ..., xn with complex coefficients. We consider the graded space of invariants in T(V) with respect to the action of G. More generally, we want to give a combinatorial interpretation for the decomposition of T(V) into irreducible representations. For the symmetric and dihedral groups, we have a link between the representations and a special subalgebra of the group algebra that gives an interpretation of T(V) in terms of words in the associated Cayley graph. In particular, we have an interpretation for the graded dimensions of the invariants in T(V) and we give closed formulas for those graded dimensions.
These examples suggest a general method to find the decomposition of T(V). To that end, we would need to find a special set of generators and relations for the group G and then the decomposition of T(V) would be linked to words in the Cayley graph of G associated to those generators.
ABSTRACT: I will present recent joint work with Jim Haglund and Jennifer Morse about the action of the operator nabla on Hall-Littlewood symmetric functions. The combinatorics of Dyck paths leads us to very surprising symmetric function identities relating Hall-Littlewood symmetric functions indexed by compositions.
ABSTRACT: A number of interesting bases exist for the upper half of the universal envelopping algebra of a semisimple Lie algebra. One such basis is Lusztig's semicanonical basis which is indexed by components of quiver varieties. Another interesting basis is indexed by Mirkovic-Vilonen cycles which lead to the combinatorics of MV polytopes. In this talk, I will explain a natural bijection between the components of quiver varieties and the MV polytopes. This is joint work with Pierre Baumann.
ABSTRACT: Many recent papers are devoted to some infinite dimensional Hopf algebras called collectively "combinatorial Hopf algebras". Among the examples we find the Faa di Bruno algebra, the Connes-Kreimer algebra and the Malvenuto-Reutenauer algebra. We give a precise definition of such an object and we provide a classification. We show that the notion of preLie algebra and of brace algebra play a key role.
ABSTRACT: We give a representation of the classical theory of multiplicative arithmetic functions (MF) in the ring of symmetric polynomials. The basis of the ring of symmetric polynomials that we use is the isobaric basis, a basis especially sensitive to the combinatorics of partitions of the integers. The representing elements are recursive sequences of Schur polynomials evaluated at subrings of the complex numbers. The multiplicative arithmetic functions are units in the Dirichlet ring of arithmetic functions, and their properties can be described locally, that is, at each prime number p. Our representation is, hence, a local representation. One such representing sequence is the sequence of generalized Fibonacci polynomials. In general the sequences consist of Schur-hook polynomials. This representation enables us to clarify and generalize classical results, e.g., the Busche-Ramanujan identity, as well as to give a richer structural description of the convolution group of multiplicative functions. It is a consequence of the representation that the MF¹s can be defined in a natural way on the negative powers of the prime p.
ABSTRACT: At first, we will have a short introduction of the diffraction theory for aperiodic point sets. Then we will move to the formulation of dynamical systems for regular model sets. Several interesting algebraical properties of regular model sets will be presented at the same time. After that, the relationship between the diffraction measures and the dynamical system measures will be discussed and it will be generalized to more general point sets, discrete point sets with positive minimum separation distance.
ABSTRACT: We have introduced a set of axioms which guarantee that the Grothendieck groups of a tower of algebras $\bigoplus_{n\ge0}A_n$ can be endowed with the structure of graded dual Hopf algebras. Hivert and Nzeutzhap, and independently Lam and Shimozono constructed dual graded graphs from primitive elements in Hopf algebras. In this paper we apply the composition of these constructions to towers of algebras. We show that if a tower $\bigoplus_{n\ge0}A_n$ gives rise to graded dual Hopf algebras then we must have $\dim(A_n)=r^nn!$ where $r = \dim(A_1)$.
This is joint with the with N. Bergeron and T. Lam.
ABSTRACT: The pre-Lie operad can be realized as a space T of labelled rooted trees. A result of F. Chapoton shows that the pre-Lie operad is a free twisted Lie algebra. That is T=(Lie o F) for some S-module F. In the context of species, we construct an explicit basis of F. This allows us to give a new proof of Chapoton's results. Moreover it permits us to show that F forms a sub nonsymmetric operad of the pre-Lie operad T.
ABSTRACT: The MacMahon master theorem is a classic theorem in enumerative combinatorics with many interesting applications. Recently, a q-right-quantum generalization was given by Garoufalidis-Le-Zeilberger. In this talk, we present a framework that not only gives a simple new proof of the classical result, but also generalizes the GLZ master theorem and gives some other extensions and applications.
ABSTRACT: There exists different ways to generalize kostka polynomials. In this talk, we will mainly focus on the ones using ribbon tableaux, crystal basis and rigged configurations. We present conjectures on the relations between these generalizations. We also present a combinatorial proof of the specialization at root of unity of Hall-Littlewood functions. Finally, we conclude on some conjectures for computing macdonald polynomials using affine quantum groups.
ABSTRACT: Fomin (1994) introduced graph duality and a generalization to dual graded graphs of the classical RSK algorithm. There are many examples of well known combinatorial Hopf algebras in which graph duality arises. We present an evidence that for any pair of dual combinatorial Hopf algebras, there is at least a dual graded graph associated. We discuss the special case of a family of sub Hopf algebras of the Malvenuto-Reutenauer algebra of permutations. Recently, Lam and Shimozono also independently obtained a construction on dual graded graphs arising from combinatorial Hopf algebras, and they generalized it to some other families of algebras including Kac-Moody algebras.
A prototypical example of dual graded graph is the Young-Fibonacci lattice which is an analogue of Young's lattice. We describe a simple mechanism to translate a chain in the Young-Fibonacci lattice into a Young-Fibonacci tableau. We investigate a relation between the combinatorics of Young-Fibonacci tableaux and the study of Okada's algebra associated to the Young-Fibonacci lattice. Indeed, we provide the set of Young-Fibonacci tableaux of size $n$ with a structure of graded poset, induced by the weak order on permutations of the symmetric group, and realized by transitive closure of elementary transformations on tableaux. We show that this poset gives a combinatorial interpretation of the coefficients in the transition matrix, from the analogue of complete symmetric functions to the analogue of Schur functions, in Okada's algebra. A similar relation is observed between usual Kostka numbers and a poset of Young tableaux studied by Melnikov and Taskin.
ABSTRACT: Determining when two skew Schur or Weyl modules over C are equivalent, or studying all binomial syzygies amongst products of skew Schur functions reduces to determining when two skew Schur functions, indexed by connected skew diagrams, are equal.
In this talk we describe some new necessary and sufficient conditions for skew Schur function equality. In particular the sufficient conditions include natural generalisations of 1) the compositions of compositions introduced by Billera, Thomas and the speaker 2) non-crossing partitions.
No prior knowledge of any of the above is required. This is joint work with Vic Reiner and Kris Shaw.
ABSTRACT: The theory of KZ equations is a generalization of the classical theory of hypergeometric functions. KZ equations arise in conformal field theory and have connections to hyperplane arrangements, representation theory, and quantum groups. A remarkable property of KZ equations is that solutions can be written in terms of multidimensional hypergeometric integrals. In this talk I will discuss two ways to construct solutions of KZ equations. Then I will show how these solutions lead to new combinatorial identities for rational functions.
ABSTRACT: Lusztig has recently extended the theory of total positivity by introducing the totally nonnegative part of a flag variety. In this talk we will describe the face poset Q^J of cells in Rietsch's cell decomposition of the totally nonnegative part of an arbitrary flag variety. Our goal is to use combinatorial techniques to understand what the space and its cell decomposition ``look like." Using tools such as Bjorner's EL-labellings and Dyer's reflection orders, we prove that Q^J is graded, thin, and EL-shellable. It follows that Q^J is Eulerian. Additionally, our results imply that the order complex of Q^J is homeomorphic to a ball, and moreover, that Q^J is the face poset of a regular CW complex homeomorphic to a ball. In particular, this resolves Postnikov's conjecture that the face poset of the totally nonnegative part of the Grassmannian is shellable and Eulerian.
ABSTRACT: Triangulations of a convex polygon are known to be counted by the Catalan numbers. A natural generalization of a triangulation is a k-triangulation, which is defined to be a maximal set of diagonals so that no k+1 of them mutually cross in their interiors. It was recently shown by Jonsson that k-triangulations are enumerated by certain determinants of Catalan numbers, that are also known to count k-tuples of non-crossing Dyck paths. However, no bijective proof is known for general k. In this talk I will present a bijection for the case k=2. The bijection is obtained by constructing isomorphic generating trees for the sets of 2-triangulations and pairs of non-crossing Dyck paths.
ABSTRACT: Kerov and Stanley introduced two different polynomials both of which evaluate characters of the symmetric group. In this talk, I will introduce these polynomials, discuss their various properties and address the main conjectures concerning these polynomials. Namely, it is conjectured that both polynomials display positivity properties. Although the positivity questions are not answered fully, some progress has been made on them.
Muge Taskin, University of Minnesota, will give a talk on "Properties of Four Partial Orders on Tableaux" at 4:00p.m. in N638 Ross.
ABSTRACT: Standard Young tableaux has been well know with their connection with the representation theory of symmetric group and special linear algebra sl[n]. In this talk we will focus on the following four partial orders which are induced from this connection: weak, KL, geometric and chain orders. After recalling their definitions and some of their crucial properties we will discuss three main results about these orders. The first one is related to the product in a Hopf algebra of tableaux defined by Poirier and Reutenauer. The second one is about the homotopy type of their proper parts. The last one addresses two of these orders which can be defined on the skew tableaux having fixed inner boundary, and similarly analyzes their homotopy type and Mobius function. The talk will further include some preliminary results about domino tableaux, which are also related to the representation theory of symplectic algebra sp[2n] and orthogonal algebra so[2n+1].
ABSTRACT: The Euclidean algebra is the Lie algebra of the group generated by the rotations and translations in the Euclidean plane. Little is known about the class of finite dimensional representations of this algebra. A classification of these reprepresentations was obtained by A. Douglas when the representations have no more than two generators. In this talk, we will discuss the class of finite dimensional representations of the Euclidean algebra which are obtained by embedding it into the Lie algebra of trace zero 3 by 3 matrices. We also give graphical descriptions for those representations in the class which have no more than two generators.
ABSTRACT: We are interested in obtaining geometric information about the solution sets to a particular class of ill-posed, discrete inverse problems. The solution sets are topological manifolds, and their structure is naturally encoded in polyhedral complexes built out of permutohedra. The required topological information is ultimately revealed by a certain quotient map that induces a curious mapping from the permutohedron to the cube. In this talk we will describe the combinatorial geometry involved and sketch its connection to the original inverse problem.
ABSTRACT: Smooth infinite words over $\Sigma =\{ 1,2\}$ are connected to the Kolakoski word $K = 221121 \cdots $, defined as the fixpoint of the function $\Delta $ that counts the length of the runs of 1's and 2's. We will survey some of the combinatorial properties shared by these words, and recall some long standing conjectures due to Dekking. The notion of smooth words extends to arbitrary alphabets and reveal some surprising combinatorial properties: it provides a new representation of the infinite Fibonacci word $F$ as an eventually periodic word. On the other hand, the Thue-Morse word is represented by a finite one.
ABSTRACT: Finitely generated fully residually free groups, also known as freely discriminated or limit groups, play an important role in the theory of equations over free groups. It turns out that many combinatorial techniques used in connection with free groups can be applied also to the class of fully residually free groups, to obtain solutions of many algorithmic problems for these groups. We show how to associate with any finitely generated fully residually free group a labeled graph of a special type, which we then use via methods similar to Stallings' "foldings technique" for free groups, to effectively solve the membership problem, the conjugacy problem, the malnormality problem, provide algorithms to compute the intersection of finitely generated subgroups, ranks of centralizers, cohomological dimension, etc.
ABSTRACT: When I started my PhD I learnt many new mathematical ideas and techniques. Perhaps the most important of these was "It is always easier to prove something when you know it is true".
I have used this idea a great deal in my work; computer aided number crunching has paved the way to many of my results. Recently I was introduced the problem of counting pattern-avoiding permutations. This problem arises in different contexts in computer science and algebra. In the last decade or so it has been subject to a great deal of work and a number of conjectures have been made. Some of these have recently been proved, but much work remains to be done.
In this talk I will give a general discussion of pattern-avoiding permutations before focussing on some of the experimental work I have done which led me to realise (but not prove) that two open conjectures were false. Thankfully it *is* easier to disprove something when you know it is false, and I will show you how we disproved one conjecture and some of our work towards disproving the other.
ABSTRACT: Abstract: Lusztig showed that all polynomials p(x_1,...,x_n) in the dual canonical basis satisfy p(A) >= 0 for every totally nonnegative matrix A = (a_ij). It was shown in 2004 that the evaluation of these polynomials at Jacobi-Trudi matrices yields Schur nonnegative symmetric functions. We will discuss variations of these properties and their relation to cluster algebras and Schubert varieties.
ABSTRACT: An nxn matrix W is type II if W^{-1})_{ij}= 1/n W_{ji} for i, j = 1, ..., n. Using Nomura's construction, each type-II matrix W yields an association scheme, also called the Nomura algebra of W.
In this talk, we will introduce type-II matrices and their Nomura algebras. We will discuss the connections of type-II matrices to various combinatorial objects.
ABSTRACT: For any subset I of {1,2,...,n-1}, let u_I denote the sum of all permutations of {1,2,...,n} with descent set I. It is well-known that Solomon's descent algebra is the linear span of the elements u_I. The fact that we have an algebra means that there exist integers c_{I,J}^K (in fact they are nonnegative) such that u_I u_J = sum_{K} c_{I,J}^K u_K. We call the c_{I,J}^K the structure constants for the algebra with respect to the basis {u_I}. The question is, how do we prove that such integers exist, and if they exist, do we have a nice (combinatorial) description for them?
While others have answered this question, we will highlight an approach that employs Stanley's P-partitions. Via quasisymmetric generating functions, it gives a simple combinatorial description to the structure constants in the Solomon descent algebra. More importantly, the same method can be used (with slight modifications) to get structure constants for several other descent-like algebras, including (so far): Solomon's type B descent algebra, the interior descent algebra (of type B), the Mantaci-Reutenauer algebra with two colors, the peak algebra, the left peak algebra, and a type B peak algebra.
ABSTRACT: Positively correlated random variables have found numerous applications in probability theory, combinatorics, and statistical mechanics through the FKG inequality, Ahlswede-Daykin theorem, and related results. The theory of negatively correlated random variables is not as well-developed although the potential applications are substantial. We'll survey some well-known theorems and conjectures considering conditions that are either necessary or sufficient for negative correlation, present a few new partial results, and indicate some lines of current research.
ABSTRACT: We formulate the combinatorics of renormalization in perturbative quantum field theory in terms of triangular matrices with entries in a Rota-Baxter algebra. A factorization of such matrices is derived using a generalization of Spitzer's identity to complete filtered non-commutative Rota-Baxter algebras. This simple matrix decomposition is used to characterize the process of renormalization in purely algebraic terms. We thereby recover a matrix representation of the Birkhoff decomposition of Connes and Kreimer found in their Hopf algebraic description of renormalization. The notion of Rota-Baxter algebra, which is an algebra with a linear endomorphism P that satisfies the relation P(x)P(y)=P(xP(y))+P(P(x)y)-qP(xy) is reviewed. Here q is a fixed constant. At the end we will describe the construction of free Rota--Baxter algebra in terms of generalized shuffles on decorated rooted trees.
ABSTRACT: The associahedron was discovered by J. Stasheff in 1963. It is a simple (n-1)-dimensional convex polytope whose 1-skeleton is given by the Tamari lattice on the set of triangulations of an (n+2)-gon.
An elegant and simple realization was recently given by J.-L. Loday: label the vertices by planar binary trees with (n+2) leaves and apply a simple algorithm on trees to obtain integer coordinates. An interesting aspect of this construction is that the permutahedron can be obtained from the associahedron by truncation.
The associahedron fits, up to combinatorial equivalences, in a larger family of polytopes discovered by S. Fomin and A. Zelevinsky. Among these generalized associahedra, the cyclohedron is a simple n-dimensional convex polytope whose vertices are given by the set centrally symmetric triangulations of a (2n+2)-gon.
It is a natural question to ask for a construction similar to Loday's for the cyclohedron. We provide such a construction in this work: First, we construct many realizations of the associahedron, each one is associated to an orientation of the Coxeter graph of type A and generalizes Loday's construction. Then we provide a realization of the cyclohedron if n is even and the orientation symmetric.
This is joint work with Carsten Lange.
ABSTRACT: We will discuss certain monomial bases of the irreducible representations of the quantum universal enveloping algebra of a Kac-Moody Lie algebra and, for the affine Kac-Moody Lie algebra of type A_n^(1), show how to compute them using the crystal graphs of the representations. We will also see that the transition matrices from these monomial bases to the Global crystal bases are upper triangular with ones in the diagonal.
ABSTRACT: The classifying spaces for the symmetric groups carry a rich combinatorial structure, reflecting the fact that they classify covering spaces. The mod 2 homology of the symmetric groups becomes a bialgebra which is generated by Kudo-Araki operations. We describe it as the free Q-ring on one generator, defined using generating function identities in the operations.
The freeness is based on a dual pairing with the Dickson algebras of GL(n) invariants. This duality also helps explain the use of vector symmetric functions in work of Adem-Milgram and Feshbach on the cohomology of symmetric groups.
ABSTRACT: The descent algebra of the symmetric group S_n, discovered by Solomon in the wider context of finite Coxeter groups, is a subalgebra of the integral group algebra of S_n. It is intimately related to the free algebra L(V) over a finite-dimensional vector space V, due to the work of Garsia and Reutenauer.
The graded component L^n(V) of L(V) is the nth Lie power of V. In the classical case (over a field of characteristic zero) the descent algebra has proven to be of tremendous help for the study of L^n(V) as a module for the general linear group GL(V). In fact, the structure of L^n(V) is reasonably well understood in this case.
The structure of modular Lie powers (over a field of prime characteristic p) has largely been a mystery. In recent collaboration with Bryant and Erdmann, we employed the descent algebra in this case as well. We could reduce the general problem to the case where n is a power of p. As a by-product, there is a set of canonical modules B_n which are determined by their characters and which can therefore be studied using the theory of symmetric functions. These modules are defined implicitly by a description of the ghost components of the Witt vector (B_1,B_2,B_3,...).
So far, we have not been able to derive an explicit description of B_n from this, and we would like to invite the audience to introduce their combinatorial and algebraic skills onto the subject. Accordingly, the lecture will be as self-contained as possible.
ABSTRACT: Associated to any finite reflection group, there is a combinatorial object called the lattice of noncrossing partitions. In type A, these are just the classical noncrossing partitions. In this talk, I will discuss a new approach to the lattice of noncrossing partitions for crystallographic reflection groups, using the representation theory of quivers (the basics of which I will explain). This approach yields a new proof that the noncrossing partitions do indeed form lattices for these groups, and also clarifies and makes precise connections between noncrossing partitions, pre-Cambrian lattices, and clusters. Time permitting, I will make a few comments about the situation for affine reflection groups, where less is known, but a similar approach looks promising. This is work in progress, joint with Colin Ingalls.
ABSTRACT: We investigate the problem of conservation of combinatorial structures in genome rearrangement scenarios. We characterize an interesting class of signed permutations, called perfect permutations, for which one can compute in polynomial time a reversal scenario that conserves all common intervals and is parsimonious among such scenarios. The general problem has been shown to be NP-hard (Figeac & Varre, 2004) We show that there exists a class of permutations, called commuting permutations, for which this computation can be done in linear time, while in the larger class of perfect permutations, the computation can be achieved in quasi-linear time. Our algorithms rely on an interesting relation between common intervals and the theory of modular decomposition of permutation graphs.
Work in collaboration with Severine Berard, Anne Bergeron and Christophe Paul.
ABSTRACT: S. Fomin and A. N. Kirillov defined certain stable Grothendieck polynomials. A combinatorial rule for the monomial expansion of these polynomials was known from the beginning. What was not understood was a generalized Littlewood-Richardson rule for the expansion in terms of the natural basis of stable Grothendieck polynomials for partitions.
I'll describe some of the motivation and known results obtained for these polynomials over the past decade. I'll then present a formula for the aforementioned expansion, developed with A. Buch, A Kresch, M. Shimozono and H. Tamvakis. Our main proof technique is an extension of the classical Edelman-Greene/Robinson-Schensted insertion algorithms.
ABSTRACT: The peak algebra is a unital subalgebra introduced by Aguiar-Bergeron-Nyman as the image of the descent algebra of type B under the map that forgets signs. A linear basis of the peak algebra is given by sums of permutations with common peak set. By exploiting the combinatorics of sparce subsets of $[n-1]$ and compositions of $n$ called almost-odd and thin, we construct three new linear bases of the peak algebra. In this talk we use the above basses to describe the Jacobson radical of the Peak algebra and to characterize the elements of the Peak algebra in terms of its action on the tensor algebra of a vector space.
Joint work with M. Aguiar and K. Nyman.
ABSTRACT: Last week in PART I: We have introduced the Hopf algebra of symmetric functions in noncommutative variables. Also we discussed the combinatorics of set partitions (lattice operations, restriction operation and the join operation). We also showed that the partition lattice algebra is commutative and semi-simple.
This week in PART II: We show that the Grothendick bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.
ABSTRACT: We show that the Grothendick bialgebra of the semi-tower of partition lattice algebras is isomorphic to the graded dual of the bialgebra of symmetric functions in noncommutative variables. In particular this isomorphism singles out a canonical new basis of the symmetric functions in noncommutative variables which would be an analogue of the Schur function basis for this bialgebra.
This is joint work with C. Hohlweg, M. Rosas and M. Zabrocki.
ABSTRACT: We define the Temperly-Lieb immanants using the Temperly-Lieb algebra and prove that these immanants are totally nonnegative, Schur nonnegative, and satisfy a natural generalization of Lindstr¬omÕs Lemma. The Temperly-Lieb immanants are also shown to characterize all totally nonnegative linear combinations of products of two complemenary matrix minors. As applications of this theory, we prove combinatorial tests for determining whether a linear combination of products of two complementary minors is totally nonnegative or Schur nonnegative and show that the dimension of the linear span of complementary minor products of an nxn matrix is the nth Catalan number, C_n. Finally, we give several generalizations of results from linear algebra using these immanants. This is joint work with Mark Skandera at Dartmouth College.
ABSTRACT: The primary focus of this talk will be the Hurwitz enumeration problem, which asks for the number H_0(\pi) of decompositions of a given permutation \pi into an ordered product of a minimal number of transpositions such that these factors act transitively on the underlying set of symbols. (The problem is typically phrased in terms of counting almost simple branched coverings of the sphere by the sphere with arbitrary ramification over one special point, but the two phrasings are equivalent.) I shall demonstrate that these transitive factorizations can be encoded as planar edge-labelled maps with certain descent structure, and describe a bijection that ``prunes trees'' from such maps. This allows for a shift in focus from the combinatorics of factorizations to the sometimes more manageable combinatorics of smooth maps. As a result, we gain combinatorial insight into the nature of Hurwitz's famous formula for H_0(\pi), and derive new bijections that prove his formula in certain restricted cases.
ABSTRACT: The multisymmetric polynomials are the diagonal invariants of the symmetric group. We will present some problems about these objects: presentation of their algebra by generators and relations and Foulkes-Howe conjecture. We will discuss the direct applications in algebra: isomorphism between the symmetric product and the Chow variety of multisets of points of the projective space, study of systems of polynomial equations in with finitely many solutions, charactacterization of the polynomials (in several variables) that factorize totally (i.-e. as products of polynomials of degree 1).
ABSTRACT: De Concini and Procesi studied a family of ideals, indexed by partitions in connection with their studies of some particular families of flag varieties. In this work we compute their resolutions, and give an explicit formula for their Poincare polynomial, when the ideals happen to be indexed by an hook shape.
This is joint work with Riccardo Biagioli and Sara Faridi.
ABSTRACT: The objects of study here are two-dimensional lattice walks, with a fixed set of step directions, restricted to the first quadrant. These walks are well studied, both in a general context of probabilistic models, and specifically as particular case studies for fixed direction sets, notably the so-called Kreweras' walks defined by the direction set {NE, W, S}.
The goal here is to examine two series associated to these walks: a simple length generating function, and a complete generating function which encodes endpoints of walks, and to determine combinatorial criteria which decide when these series are algebraic, D-finite, or none of the above. (Indeed we have examples that we believe to be non-D-finite) We shall present an (almost) complete classification of all nearest neighbour walks where the set of directions is of cardinality three, and discuss how this leads to a natural, well supported, conjecture for the classification of nearest walks with any direction set.
Work in progress with M. Bousquet-Melou.
ABSTRACT: Let L be a finite lattice, that is, L is a finite set equipped with a partial order, such that every pair of elements x,y has a least upper bound x v y and a greatest lower bound xy . A real valued function f on L is said to be supermodular if for all x,y in
L f(x v y) + f(xy) >= f(x) + f(y)If equality holds for all x and y, f is said to be modular
The problem we consider is that of determining the extreme rays of the quotient S/M where S is the cone of supermodular functions, and M is the vector space of modular functions. ( A ray R in a cone K is said to be extreme if a =b+c where a is in R and b,c are in K implies that b,c are in R.)
This was motivated by problems in probability theory dealing with stochastic orderings.( The particular application is to determine dominance for the supermodular ordering on multivariate distributions).
We are able to solve this problem completely for the simplest type of lattices, namely those which are the disjoint union of chains.
For the particular application, we are interested in the lattice Z_N ^k consisting of all k -tuples with entries from the set {0,1, ... N-1}, equipped with the usual pointwise order. We are able to get complete answers for the cases : k = 2; N=2 , k= 3 or 4; N=3, k = 3. We present some conjectures for the general case.
ABSTRACT: Recently I emailed Nantel a short applied algebra question (asked jointly with Dylan Thurston), and he (along with Christophe Hohlweg) quickly solved it. In return, I have to explain why is it interesting.
After quickly stating the question I'll tell you about categorification (a bold suggestion of I. Frenkel, that much of math is the Euler characteristic of some "higher math", much like much of algebra is q-algebra at q=1). I'll then define traces and trace groups, which allow Euler characteristics to take values in objects more interesting than merely numbers. Finally I'll introduce the category of matrix factorizations, which is the core of a surpising new method for constructing homological theories from local data.
The ideas to be introduced in my talk (categorifcation, trace groups and matrix factorizations) are all conceptual and foundational and worthy of your time, definitely more than the incomplete (though possibly valid) logic that lead us to our question to Nantel. So assuming some luck, I'll only have time to tell the latter part of the story over coffee after my talk. I hope there's good coffee up there north of 401.
ABSTRACT: Louis Solomon showed that the group algebra of the symmetric group has a subalgebra called the descent algebra, generated by sums of permutations with a given descent set. For any Weyl group, Paola Cellini proved the existence of a different, commutative subalgebra of the group algebra. We derive the existence of such a commutative subalgebra for the case of the symmetric group and of the hyperoctahedral group using a variation on Richard Stanley's theory of $P$-partitions.
ABSTRACT: In Part I we have introduced the theory of languages and finite automata. We talked about the Chomsky heirarchy of langauages and discuss the relationship between classes of languages and their generating functions.
Part II: Classes of permutations which do not contain certain patterns arise in several areas of algebraic combinatorics (e.g. in Schubert calculus vexillary permutations avoid the pattern 2143). The problem of enumerating permutations which avoid any given pattern has become a small subfield of combinatorics in recent years. There are very few general techniques to approach these enumeration problems and we will discuss a new way of looking at pattern avoiding permutations which will help to understand some of the results that are known.
In the second part we will apply last week results to pattern avoiding permutations and show that the class of permutations with a fixed number of descents that avoid any fixed pattern has a rational generating function.
A permutation, pi in S_n, is said to contain a pattern, X in S_m with m<=n, if there is a subword of pi, pi_{i_1} p_{i_2} ... p_{i_m} where i_r < i_{r+1} such that pi_{i_r} < pi_{i_s} if and only if X_r < X_s. e.g. 6132754 contains the pattern 4132 since the subword 6254 has the same relative order 6132754 avoids the pattern 1234 since there is no increasing subsequence of 4 numbers.
ABSTRACT: Classes of permutations which do not contain certain patterns arise in several areas of algebraic combinatorics (e.g. in Schubert calculus vexillary permutations avoid the pattern 2143). The problem of enumerating permutations which avoid any given pattern has become a small subfield of combinatorics in recent years. There are very few general techniques to approach these enumeration problems and we will discuss a new way of looking at pattern avoiding permutations which will help to understand some of the results that are known.
This talk will include an introduction to the theory of languages and finite automata. We will talk about the Chomsky heirarchy of langauages and discuss the relationship between classes of languages and their generating functions. Finally, we will apply these results to pattern avoiding permutations and show that the class of permutations with a fixed number of descents that avoid any fixed pattern has a rational generating function.
A permutation, pi in S_n, is said to contain a pattern, X in S_m with m<=n, if there is a subword of pi, pi_{i_1} p_{i_2} ... p_{i_m} where i_r < i_{r+1} such that pi_{i_r} < pi_{i_s} if and only if X_r < X_s. e.g. 6132754 contains the pattern 4132 since the subword 6254 has the same relative order 6132754 avoids the pattern 1234 since there is no increasing subsequence of 4 numbers
ABSTRACT: The descent algebra has been introduced by Solomon, with an homomorphism which takes values on the algebra of characters, to solve a problem on characters theory of finite Coxeter groups.
We introduce two new properties of this homomorphism and give applications on characters theory.
ABSTRACT: A crystal for a representation of a semisimple Lie algebra is a combinatorial object which encodes the structure of the representation. There is an interesting tensor product on these crystals. We give a construction of a commutor (natural isomophisms A x B -> B x A) for the category of crystals of a semisimple Lie algebra. This commutor is symmetric but does not satisfy the usual hexagon axiom. Instead it obeys a different axiom which makes the category of crystals into a coboundary category. Motivated by the above construction, we investigate the structure of coboundary categories. Just as the braid group acts on repeated tensor products in a braided category, the fundamental group of the moduli space of stable real genus 0 curves with n marked points acts on repeated tensor products in a coboundary category.
ABSTRACT: Andrews and Curtis conjectured in 1965 that every balanced presentation (i.e. with an equal number of generators and relations) of the trivial group can be transformed into a standard presentation by a finite sequence of elementary transformations. Their conjecture was originally motivated by topological questions. Some recent computational work by Miasnikov and Myasnikov on this problem is based on genetic algorithms. We show that a computational attack based on a breadth-first search of the tree of equivalent presentations is also viable, and seems to outperform that based on genetic algorithms. It allows us to extract shorter proofs (in some cases, provably shortest). We discuss short potential counterexamples.
The talk will be accessible to a fairly wide audience of those interested in algebra or computer science.
ABSTRACT: I will describe some recent work that aims to generalize a well-known lower central series formula for the fundamental group of a hyperplane arrangement complement. The main object of interest here is a "homotopy Lie algebra" which arises in commutative algebra and rational homotopy theory. I will show that one can describe the algebra explicitly in terms of combinatorial ingredients in certain cases, then give some applications.
ABSTRACT: The work of Koike, Terada and Kleber defines three distinguished bases of the symmetric functions related to the characters of the Lie groups of type $B_n$, $C_n$ and $D_n$. We give formulas for these bases and find a uniform way to express them and manipulate them. Using techniques developed to analyze Hall-Littlewood and Macdonald symmetric functions we consider the q-deformation of these bases and find that they are related to the X=M conjecture of Hatayama, et al. allowing us to give a formula for coefficients K which we conjecture are equal to X and M (X=M=K conjecture).
This is joint work with Mark Shimozono.
ABSTRACT: We introduce a noncommutative binary operation on matroids, called the free product, and discuss some of its properties. In particular, free product is characterized by a certain universal property, is associative, and respects matroid duality. We characterize matroids that are irreducible with respect to free product and show that, up to isomorphism, every matroid factors uniquely as a free product of such matroids. We use these results to prove an inequality involving the numbers of nonisomorphic matroids on n elements which was conjectured by Welsh, and to show that the Hopf algebra of matroids with restriction-contraction coproduct is cofree.
ABSTRACT: Studies on the combinatorics of descents in permutations led to the discovery of a pair, (QSym,NSym), of mutually dual graded Hopf algebras. Here, QSym is the graded Hopf algebra of quasi-symmetric functions, and its graded dual, NSym, is the graded Hopf algebra of noncommutative symmetric functions. Recent investigations on the combinatorics of peaks in permutations resulted in the discovery of an interesting new pair, (Peak,Peak*), of graded Hopf algebras. The first one, Peak, originally due to Stembridge, is a subalgebra of $QSym$. Its graded dual, Peak*, can therefore be identified as a homomorphic image of NSym. Our main result is to provide a representation theoretical interpretation of (Peak,Peak*) as Grothendieck rings of the tower of Hecke-Clifford algebras at $q=0$.
This is joint work with F. Hivert and J.Y. Thibon (Marne la vallee, France).
ABSTRACT: A group is Hopfian if each surjective endomorphism of the group has trivial kernel. Zlil Sela proved that torsion-free hyperbolic groups are Hopfian. His proof is based on properties of JSJ decompositions of groups, Rips' theory of groups acting on real trees and theorems due to Bestvina and Paulin on degeneration of hyperbolic structures. I will explain how Sela's proof works and why it extends to subgroups of torsion-free hyperbolic groups. If time permits, I will also explain how the techniques used in this proof apply to algebraic geometry over groups.
ABSTRACT: Steinberg unitary Lie algebras were introduced by Allison and Faulkner which is a unitary version of Steinberg Lie algebras. We will use twisted vertex operators over Fock space to construct modules for a class of Steinberg unitary Lie algebras. This is a joint work with Naihuan Jing.
ABSTRACT: Given a partially ordered set (poset) P and a labelling of its vertices, we will give a definition of a P-partition, as introduced by Richard Stanley in his Ph.D. thesis. In this thesis, Stanley made a conjecture concerning a certain quasi-symmetric generating function for the set of P-partitions of a labelled poset. This conjecture, which remains open, says that the generating function is a symmetric function if and only if our labelled poset is a "Schur labelled skew shape poset." In 1995, Claudia Malvenuto reformulated the conjecture so that the symmetry of the generating function needs to be related only to the local structure of the labelled poset, rather than its global structure. We will discuss a generalization of the idea of a P-partition, an appropriate extension of Stanley's conjecture, and an extension of Malvenuto's reformulation. We will also explain how Stanley's conjecture is almost always true and discuss several open problems concerning these quasi-symmetric generating functions.
ABSTRACT: The talk will begin with a survey of some of the main enumerative results in the subject of restricted (or pattern-avoiding) permutations. Next, recent developments and new directions will be discussed, including simultaneous avoidance of several patterns, enumeration of occurrences of a particular pattern in permutations, and generalized patterns (i.e., with the requirement that some elements occur in adjacent positions). The second part of the talk will focus on the study of statistics in restricted permutations, in which bijections to Dyck paths play an important role. We give a new bijection between 321-avoiding permutations and 132-avoiding permutations that preserves the number of fixed points and the number of excedances. We will discuss recent work with Emeric Deutsch, Toufik Mansour, Marc Noy and Igor Pak.
ABSTRACT: The Kostka numbers $K_{\lambda\mu}$ appear in combinatorics when expressing the Schur functions in terms of the monomial symmetric functions, as $K_{\lambda\mu}$ counts the number of semistandard Young tableaux of shape $\lambda$ and content $\mu$. They also appear in representation theory as the multiplicities of weights in the irreducible representations of type $A$.
Using a variety of tools from representation theory (Gelfand-Tsetlin diagrams), convex geometry (vector partition functions), symplectic geometry (Duistermaat-Heckman measure) and combinatorics (hyperplane arrangements), we show that the Kostka numbers are given by polynomials in the cells of a complex of cones. For fixed $\lambda$, the nonzero $K_{\lambda\mu}$ consist of the lattice points inside a permutahedron. By relating the complex of cones to a family of hyperplane arrangements, we provide an explanation for why the polynomials giving the Kostka numbers exhibit interesting factorization patterns in the boundary regions of the permutahedron. We will consider $A_2$ and $A_3$ (partitions with at most three and four parts) as running examples, with lots of pictures.
I will also say a few words as to how some of the techniques used
generalize to the case of Littlewood-Richardson coefficients.
This is joint work with Sara Billey and Victor Guillemin.
ABSTRACT: Fomin, Fulton, Li, and Poon defined an operation on pairs of partitions. They conjectured that a certain symmetric function associated to this operation is Schur-positive. We obtain a combinatorial description of this operation that allows us to generalize the conjecture, to prove many instances of it, and to show that it holds asymptotically.
This is join work with Francois Bergeron and Riccardo Biagioli.
ABSTRACT: I will report on some joint work with B. Allison and A. Pianzola in which we investigate iterated loop algebras and obtain a general ``permanance of base '' result. Using this we show that the ``type'' and ``number of steps '' of an iterated loop algebra of a split simple Lie algebra are isomorphism invariants.
ABSTRACT: We find that a Fock space which is the basic module for the affine general Lie algebra $\hat{gl}__N$ allows both the quantum affine general algebra $U_q(\hat{gl}__N)$ and the quantum toroidal algebra actions by using the vertex operator construction. As a byproduct, we enhance a Lusztig theorem on quantum affine algebras.
ABSTRACT: There are several connections between permutation statistics on the symmetric group and the representation theory of the symmetric group S_n. After giving a brief survey of the known results for S_n, I will show how to generalize them to the even-signed permutation group D_n. In particular I will define a major index and a descent number on D_n that allow me to give an explicit formula for the Hilbert series for the invariants of the diagonal action of D_n on the polynomial ring. Moreover I will give a monomial basis for the coinvariant algebra R(D). This new basis leads to the definition of a new family of D_n modules that decompose R(D). An explicit decomposition of these representations into irreducible components is obtained by extending the major index on particular standard Young bitableaux.
This is a joint work with Fabrizio Caselli.
ABSTRACT: We will introduce the notion of characters for graded Hopf algebras and derive some interesting properties of the character group. In particular we will give a canonical factorization of a character into an even and an odd character. We will discuss some canonical character related to combinatorial Hopf algebras.
ABSTRACT: We look at the relationship between a finite set of points and its ideal. One way of studying this is by the use of Hilbert functions, which is a sequence of numbers that tells us the number of independent forms of each degree that pass through the given points. A special kind of ideal, known as lex-segment ideals, satisfy many extremal properties among all ideals with the same Hilbert function. As a result, many people have looked into ways of generalizing lex-segment ideals in ways one can still be left with some of the extremal properties. Lex plus powers ideals are one such generalization. The ideal is split into two parts: the lex part and the "powers" part. The powers part defines a maximal length regular sequence that the ideal contains, and it is conjectured that lex plus powers ideals satisfy extremal properties among all ideals containing a regular sequence in the same degrees and having the same Hilbert function.In this talk, I will provide some evidence to believe this conjecture.
ABSTRACT: We characterize the class of finite solvable groups by two-variable identities in a way similar to characterization of finite nilpotent groups by Engel identities. More precisely, a sequence of words $u_1,\ldots,u_n,\ldots $ is called correct if $u_k\equiv 1$ in a group $G$ implies $u_m\equiv 1$ in a group $G$ for all $m>k$.
We are looking for an explicit correct sequence of words $u_1(x,y),\ldots,u_n(x,y),\ldots$ such that a group $G$ is solvable if and only if for some $n$ the word $u_n$ is an identity in $G$.
Let $u_1=x^{-2}y^{-1} x$, and $u_{n+1} = [xu_nx^{-1},yu_ny^{-1}]$.
The main result states that a finite group $G$ is solvable if and only if for some $n$ the identity $u_n(x,y)\equiv 1$ holds in $G$.
From the point of view of profinite groups this result implies that the provariety of prosolvable groups is determined by a single explicit proidentity in two variables.
The proof of the main theorem relies on reduction to Thompson's list of minimal non-solvable simple groups, on extensive use of the arithmetic geometry (Lang - Weil bounds, Deligne's machinery, estimates of Betti numbers, etc.) and on application of computer algebra and geometry (SINGULAR, MAGMA).
Joint work with: T.Bandman, G.-M.Greuel, F.Grunewald, B.Kunyavskii, G.Pfister
ABSTRACT: A law u(x_1,...,x_n)=u(x_1,...,x_n) is called positive if the words u and v do not contain inverses of variables, e.g. xy=yx.
A group G is called locally graded if every finitely generated subgroup in G has a nontrivial finite image.
We show equivalence of some known problems and give an affirmative answer in the class of locally graded groups.
Consider the algebra of formal power series in countable many noncommuting variables over the rationals. The subalgebra \Pi(x) of symmetric functions in noncommuting variables consists of all elements invariant under permutations of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, and complete homogeneous, and Schur symmetric functions as well as investigating their properties. A preprint can be downloaded at: www.ma.usb.ve/~mrosas/articles/articulos.html.
ABSTRACT: The cardinality of a set of certain directed bipartite graphs is equal to that of a set of permutations that satisfy certain criteria. In this talk we decribe both the set of graphs and the set of permutations, what their cardinality has to do with the chromatic polynomial, and whether we can find a natural bijection between the sets.
This is joint work with Richard Ehrenborg.
ABSTRACT: In this talk we associate to a simplicial complex a
square-free monomial ideal called its "facet ideal". We use the combinatorial
structure of a simplicial complex to study algebraic properties of its
facet ideal. We explore dualities with Stanley-Reisner theory, and describe
effective ways to compare the Stanley-Reisner complex to the facet complex
for a given monomial ideal. As a result, we re-interpret some existing
Cohen-Macaulay criteria, and produce some new ones.
ABSTRACT: The coefficients in a q-binomial coefficient mod (q^n-1) are shown
to be equal to the number of double cosets C_n\S_n/S_k \times S_{n-k}
whose size is certain multiple. This extends an observation of Chapoton. The
result is generalized to finite Coxeter groups. q-analogues are conjectured
which generalize results of Andrew and Haiman. A related Schur function result
is given.
ABSTRACT: We consider, on the space of polynomials in $2 n$ variables
$X=x_1,\ldots,x_n$ and $Y=y_1,\ldots,y_n$, the usual action of the group
$S_n\times S_n$. Using a classical result of Steinberg, this space $Q[X,Y]$
can be viewed as a $n!^2$ dimensional module over the invariants of the
group. This is to say that polynomials in X and Y can be uniquely decomposed
as linear expressions in covariants, with coefficients that are invariants.
We uses theses results, together with restriction to $S_n$ (considered
as a diagonal subgroup), to decompose diagonal alternants. In particular,
we give an explicit basis for diagonal alternants, modulo the ideal generated
by products of symmetric polynomials in $X$ and $Y$. The construction of
this basis involves a very nice classification of configurations on $n$
points in $R^2$.
ABSTRACT: I will discuss two recent results about the complex
of injective words, both of which are joint work with Phil Hanlon.
The first is a Hodge-type decomposition for the S_n-module structure for
the (top) homology, refining a recent formula of Reiner and Webb.
A key ingredient was to show that the Eulerian idempotents interact in
a nice fashion with the simplicial boundary map on the complex of injective
words. The second result deals with a recent conjecture of Uyemura-Reyes,
namely that the random-to-random shuffle operator has integral spectrum.
We prove that this conjecture would imply that the Laplacian on each chain
group in the complex of injective words also has integral spectrum.
ABSTRACT: The generalized symmetric group G(n,m) may be viewed
as the group of square matrices of size n having one non-zero entry in
each row and column, this entry being a m-th root of unity. We will talk
about two action of the (generalized) symmetric group on polynomials (symmetrizing
and quasi-symmetrizing action) and will study their invariants and covariants
(quotient by invariants). We will make and important use of Grobner bases.
ABSTRACT: The Clifford (or Weyl) algebras have natural representations
on the exterior (or symmetric) algebras of polynomials over half of generators.
Those representations are also important in quantum and statistical mechanics
where the generators are interpreted as operators which create or annihilate
particles and satisfy Fermi (or Bose) statistics. In this talk, I will
present a model for the extended affine Lie algebra over quantum tori via
the fermionic (or bosonic) construction.
ABSTRACT: Several of the past talks in our seminar were concerned
with Descent Algebras, Quasi-symmetric functions algebra as well as Peak
functions algebra..
We will discuss the links among these algebras, in particular see that the peak functions algebras is closely related to the descent algebras of type B.
Steve Berman, University of Saskatchewan, will give the first talk
which is entitled "Covering Algebras of Kac-Moody Lie Algebras and EALA's"
at 3:30p.m. in N638 Ross.
ABSTRACT: In joint work with B. Allison and A. Pianzola we have developed a theory of covering algebras. We can obtain new EALA's from old ones using this theory as well as obtaining a vast generalization of V. Kac's early work on covering algebras of the finite dimensional simple Lie algberas. This theory, together with applications, will be presented in this talk.
ABSTRACT: Let Hom(V) be the set of quivers V_0 -> V_1 -> ... -> V_n. A quiver cycle is a subset O_r of Hom(V) where the ranks of the composite maps V_i -> V_j are bounded above by specified integers r=(r_{ij}) for i < j. Our goal is to compute the equivariant cohomology class [O_r]. As a special case one obtains Fulton's universal Schubert polynomials. Buch and Fulton expressed [O_r] in terms of Schur functions, and conjectured a combinatorial formula for the coefficients. In particular, they conjectured that the coefficients, which directly generalize the Littlewood-Richardson coefficients, are positive. In this ongoing project, we construct a flat family whose general fiber is isomorphic to O_r, and whose special fiber has components that are direct products of matrix Schubert varieties. This proves that [O_r] is a sum of products of Stanley symmetric functions (stable double Schubert polynomials) where each summand is indexed by a list w of permutations. Our formula is obviously positive for geometric reasons and immediately implies the positivity of the Buch-Fulton formula. We conjecture that the special fiber is generically reduced, so that each list of permutations w occurs with multiplicity 1. We propose a simple nonrecursive combinatorial characterization of which lists w appear. This is joint work with Allan Knutson and Ezra Miller.
ABSTRACT: Let f be an irreducible polynomial with integer coefficients.
We consider the problem of finding primes p for which the reduction of
f modulo p is also irreducible. This problem goes back to Frobenius
and is a Chebotarev theorem in disguise.
ABSTRACT: A picture of the complete graph K_n consists of n labelled
points in the plane, connected with (n choose 2) lines. I'm going to talk
about the algebraic relations that must hold among the slopes of these
lines. This sounds like a problem in classical geometry, but it turns out
that the tools to attack it come from combinatorics. First, the equations
defining a picture can be described using the theory of combinatorial rigidity
of graphs. Second, once one knows what these equations are, one can apply
another combinatorial idea, the theory of Stanley-Reisner rings, to obtain
geometric invariants of the space of all solutions. Finally, various sorts
of labelled trees play important roles in describing these invariants combinatorially.
ABSTRACT: Let a(t) be a polynomial which has positive integer
coefficients, a constant term of one, and only real zeros. We show that
a(t) appears in the numerator of the Hilbert series of some Cohen-Macaulay
ring, and present some evidence in favor of the stronger conjecture that
a(t) is the f-polynomial of a simplicial complex. This is joint work with
Jason Bell.
ABSTRACT: Once more, the peak phenomenon! The peak set of a permutation
\pi in the symmetric group S_n consists of all 1
In our self-contained approach, there is a particular interest in inner products in P_n, arising from the ordinary multiplication of permutations. Peak counterparts of several results on the descent algebra D_n will be presented, based on the fact that P_n turns out to be a left ideal of D_n. This includes combinatorial and algebraic characterizations of P_n, the basics of peak Lie idempotents, and a number of observations on the structure of P_n and some sub-algebras.
Enough information will be provided to transfer these results to the setting of Stembridge's peak algebra, by duality.
Anthony Bonato, Wilfrid Laurier University, will speak on "The colouring order and the natural order on retracts" at 4:30p.m. in N638 Ross.
ABSTRACT: If G and H are graphs, then G \preceq H if there is an edge-preserving vertex-mapping, or homomorphism, from G to H. The quasi-order relation \preceq on the class of finite graphs gives rise to an order relation in a natural way, calledthe colouring order, written C. The order C has many intriguing properties; for example, Hedrlin proved in the 1960's that it is universal: every countable order embeds in C as a suborder. Hedrlin's proof is long and makes heavy use of category theory. In a recent tour de force, Nesetril discovered a shorter combinatorial proof of Hedrlin's theorem.
A retract of a graph is an endomorphism f that is idempotent: f^2=f. Retracts of graphs and other structures have been widely studied in semigroup theory, going back to Howie's pioneering work on the idempotents of the full transformation semigroup. The natural order on retracts is defined by f \le g if fg=gf=f. While the natural order is a familiar tool in algebraic semigroup theory, it has only recently attracted the attention of the graph homomorphism community. After giving some background on the colouring order C and the natural order, we will see how these two orders are related. In particular, we will prove that the natural order on the retracts of the infinite random graph embeds C and so is universal.
Samuel Hsiao, Cornell University, will speak on "The peak Hopf algebra and some connections to enumeration in posets" at 4:30p.m. in N638 Ross.
ABSTRACT: The peak algebra $\Pi$ was introduced by J. Stembridge in his development of enriched $P$-partitions. It is a Hopf subalgebra of the quasisymmetric functions $\Qsym.$ The Hopf structure has become important in connecting $\Pi$ to the enumeration of chains in Eulerian posets.
We will describe the structure of $\Pi,$ showing it to be a free polynomial algebra, a cofree graded coalgebra, and a free module over Schur's $Q$-function algebra. These results mirror results on the structure of $\Qsym$ and its relationship to the symmetric functions. We introduce a new basis of {\em monomial peak functions} for $\Pi$ which behaves much like the monomial basis for $\Qsym.$ For example, the stucture constants relative to this new basis count quasi-shuffles of {\em peak compositions.}
By duality, our results have implications for the algebra of chain-enumeration functionals on Eulerian posets. Earlier joint work with L. Billera and S. van Willigenburg identified the ${\bf cd}$-index as the dual basis to Stembridge's basis of fundamental peak functions. Here we find that the monomial basis for $\Pi$ is dual to N. Reading's Charney-Davis index, which appears to be an Eulerian analog of the flag $f$-vector.
A knowledge of quasisymmetric functions and flag $f$-vectors is useful but not required for this talk.
ABSTRACT: Fulton's Universal Schubert polynomials represent
general
degeneracy loci for maps of vector bundles with rank conditions coming
from a permutation. The Buch-Fulton Quiver formula expresses this polynomial
as an integer linear combination of products of Schur polynomials in the
differences of the bundles. We present a positive combinatorial formula
for the coefficients. Our formula counts sequences of semi-standard Young
tableaux satisfying certain conditions.
This is joint work with Anders Buch, Andrew Kresch and Harry Tamvakis.
Vasiliy Bludov, Irkutsk State University, Russia, will speak on "Investigations in Algebra and Discrete Mathematics at Irkutsk State University" at 4:00p.m. in N638 Ross.
Marcelo Aguiar, Texas A&M, will give a talk on "Loday's types of algebras and Rota's types of operators" at 3:00p.m. in N638 Ross.
ABSTRACT: After a quick overview of Loday's new "types of algebras", I will concentrate on the most recently defined "dendriform trialgebras" (there will be more scary names than this, but we will only be doing elementary algebra throughout). The free such trialgebra is based on the space of all rooted trees (Loday and Ronco). I will then describe several examples of trialgebras, all with a combinatorial flavor, and a general construction, which links trialgebras to one of Rota's favorite notions: Baxter operators. The punch line of the talk will involve quasi-symmetric functions, for a change.
Mohammad Rza R. Moghaddam, Ferdowsi University of Mashhad, Iran, will speak on "Varietal Multiplier of Groups" at 4:00p.m. in N638 Ross.
ABSTRACT: In this talk we introduce the notion of verbal and marginal subgroups of a given group, with respect to a variety of groups. Also the concept of varietal Schur-multiplier of a group will be defined and then we discuss some of their properties.
ABSTRACT: In this talk we will review the definition of crystal bases of an irreducible module of a symmetrizable Kac-Moody Lie algebra and show how they can be used to answer questions about the representation theory of this algebra. We will talk about describing these bases in terms of Young Tableaux.
ABSTRACT: A problem that arises from cryptography is to study the primality of the number of points on an elliptic curve over a finite field. We shall show how sieve methods give some information on this problem. We will begin by describing the cryptographic motivation and background.
ABSTRACT: After a brief tour of the development of the Nullstellensatz through David Hilbert in the 1890s and Emmy Noether in the 1920s, we follow up on a fairly recent idea by Bill Lawvere and exhibit the categorical essence of the Nullstellensatz via Birkhoff's Subdirect Representation Theorem for general algebras. Hence, we shall prove this theorem for quite general categories without requiring any particular expertise by the audience.
ABSTRACT: In 1989, Jones used spin models to construct link invariants and braid group representations. In particular, the Jones polynomial can be obtained from the simplest spin model called the Potts model. In 1995, Bannai and Bannai constructed four-weight spin models, which are generalization of spin models that give invariants for oriented links.
In an important paper by Jaeger, he discovered the first connection between spin models and Bose-Mesner algebras. In 1995, Jaeger and Nomura showed that there is a Bose-Mesner algebra attached to every spin model. A similar result for four-weight spin models also holds. Bose-Mesner algebras, which are equivalent to association schemes, have connections to a vast number of combinatorial objects such as designs, codes and distance regular graphs. Hence Jaeger's result draws the attention of combinatorialists to spin models.
Let $\schur$ denote the Schur product of two matrices, that is, $(A \schur B)_{ij} = A_{ij} B_{ij}$. Given a matrix $C$ in $M_n({\Bbb C})$, we define endomorphisms $X_C$ and $\Delta_C$ on $M_n({\Bbb C})$ by \begin{equation*} X_C(M) = CM, \quad \Delta_C(M) = C \schur M. \end{equation*} A pair of matrices $(A,B)$ from $M_n({\Bbb C})$ is called a Jones pair if $X_A$ and $\Delta_B$ are invertible and they give a representation of braid groups using Jones' construction. (We save the details for the talk.)
Jones pairs provide a generalization of both spin models and four-weight spin models. Moreover, Jones pairs also have Bose-Mesner algebras attached to them. In this talk, we give an introduction of Jones pairs and discuss the Bose-Mesner algebras associated with spin models, four-weight spin models and Jones pairs. In particular, we describe a family of five Bose-Mesner algebras attached to each Jones pair.
This is joint work with Chris Godsil and Akihiro Munemasa.
ABSTRACT: The higher Bruhat orders B(n,d) were introduced by Manin and Schechtman as generalizations of weak Bruhat order: B(n,1) is weak Bruhat order on the symmetric group S_n, and B(n,0) is the Boolean lattice Q_n. The higher Stasheff-Tamari posets are defined as partial orders on the set of triangulations of a cyclic polytope. There are reformulations of these families of posets which make the connections between them more obvious: a convex-geometric reformulation of the higher Bruhat orders due to Kapranov and Voevodsky, and a combinatorial reformulation of the Stasheff-Tamari posets due to myself. I will discuss various maps between higher Stasheff-Tamari posets and higher Bruhat orders which specialize to familar maps between S_n and Q_{n-1}, and from S_n to planar binary trees.
ABSTRACT: A number of combinatorial objects--labeled trees, allowable pairs of input-output permutations for priority queues, factorizations of an n-cycle into transpositions, and parking functions--are all enumerated by the same formula: (n+1)^(n-1). A series of related bijections have been constructed between two or more of these. Here we introduce and prove a direct bijection between priority queue allowable pairs and labeled trees that has an additional property not present in previous direct bijections: our bijection maps increasing sequences in the output permutation of the priority queue allowable pair to leaves in the tree. This gives us a full understanding of the underlying tree structure of priority queue allowable pairs. For instance, we could use this understanding to construct the analogue of a Prufer code for allowable pairs. This is joint work with A. Yong.
ABSTRACT: We define the Hilbert function of a finite set of points
in projective n-space. We then state the result of Geramita, Maroscia and
Roberts which characterizes the Hilbert functions of points in terms of
Macaulay's O-sequences. For every possible Hilbert function, they construct
a set of points having that Hilbert function. Their constructions, called
k-configurations, have several properties of interest in themselves. For
example, their Hilbert function, minimal free resolution and the degrees
of each point are easily determined. We generalize k-configurations to
constructions which preserve these properties. Furthermore, since O-sequences
are defined only in terms of the binomial coefficients, we are able to
use a generalized Pascal's triangle to generalize Macaulay's O-sequences.
We use this generalization to characterize the Hilbert functions of our
genereralized k-configurations.
Robert Aboolian, University of Toronto, will give a talk entitled
"Spatial Interaction Location Models with Exponential Expenditure Function"
from 10:30a.m. to 11:30a.m. in N638 Ross.
ABSTRACT: In this paper, we introduce a new model for locating competitive facilities, which captures key aspects of competitive facility location models such as market expansion and market cannibalization, while retaining computational solvability that allows for realistic-size applications. The model presents a direct generalization for the class of traditional spatial interaction models .
The new model is formulated as an integer program with a ''nasty'' non-linear objective function. We develop a novel approach to solve problem by using the TLP linear approximation that yields an efficient piece-wise linear approximation within a specified relative error bound. We develop a tight worse case error bound for the greedy heuristic, which is somewhat unexpected in view of the non-linearity of the underlying model.
We also develop efficient computational approaches - both exact (Branch and Bound) and approximate ( optimal with controllable error bound ), allowing for solution of fairly large-scale models.
Refreshments will be served in N620 Ross at 10:00a.m.
ABSTRACT: The Wishart distribution is the distribution of the
estimate of the covariance parameter in the multivariate Gaussian model.
In many statistical problems, the test statistic is a fairly complicated
function of this estimate and it is impossible to find its exact distribution.
This distribution therefore has to be approximated using the moments and
inverse moments of the Wishart distribution.
The Wishart distribution can be defined generally on symmetric cones. We donote by K the orthogonal subgroup of the group of automorphisms of the cone. We will show how, in this general case, all K-invariant moments for the Wishart and its inverse can be found using spherical polynomials.
In the particular case of the cone of Hermitian matrices, all moments (not necessarily K-invariant) can be obtained using the structure of the symmetric group.
ABSTRACT: I will discuss a number of classical properties of
the Schur function basis revealing the importance of combinatorics in symmetric
function theory. I will then consider a filtration of the symmetric function
space, and introduce new symmetric functions appearing, from their combinatorial
properties, to be the Schur functions of the subspaces associated to this
filtration. I will finish by discussing the connection between these new
symmetric functions and Macdonald polynomials.
ABSTRACT: The Good-Lagrange formula for the inversion of multivariated
formal power series is a central tool in the field of enumeration of tree-like
structures. In this talk, we will present an elegant bijective proof of
this formula and some application to the random generation of some families
of tree-like structures.
Joint work with Michel Bousquet, Gilbert Labelle and Pierre Leroux, LaCIM.
ABSTRACT: Last September we have discussed: Catalan number that
classically enumerate Dyck paths, and investigate the quotient ring $R_n$
of the ring of polynomials $\Q[x_1,x_2,\ldots,x_n]$ over the the ideal
generated by non-constant quasi-symmetric polynomials. We expected the
dimension of $R_n$ to be the $n$th Catalan number. Now we can prove it
all, and say even more...
ABSTRACT: We explain what is a pre-Lie algebra, where this notion
comes from and try to show why it may be of interest. We will give examples
of combinatorial and algebraic constructions of pre-Lie algebras.
ABSTRACT: Let n be a positive integer, w a group word. Consider
the class of all groups G satisfying the identity w^n=1 and having the
verbal subgroup w(G) locally finite. We show that in many cases this is
a variety.
All are Welcome!
Neil Roger, York University, will give a talk entitled "Measure
and Category" at 3:00p.m. in N638 Ross.
ABSTRACT: This talk introduces the concepts of measurability, first category and the Continuum Hypothesis, and presents some theorems combining these ideas. In particular, some peculiar sets of real numbers such as the Bernstein, Luzin and Sierpinski sets will be covered.
Seminar requirement for Masters students
Reminder: Master's Mathematics students are expected to attend the
talks of other students. Documented evidence at 6 such talks is expected.
Attendance sheets can be picked up from N519 Ross.
ABSTRACT: In an isomorphic copy of the ring of symmetric polynomials,
that is, isobaric polynomials, we introduce some families indexed by rational
weight vectors. We shall focus on constructing the rational roots of these
polynomials under a product induced by convolution. A direct application
is to the description of the roots of certain multiplicative arithmetic
functions (the "core" group) under the usual convolution product.
This is joint work with Trueman MacHenry.
ABSTRACT: I will show that the 2-extended Lie algebras in the
sense of Saito will be realized from tubular algebras, one kind of finite
dimensional algebras, via derived categories, one kind of triangular categories.
This is joint work with Geanina Tudose.
ABSTRACT: Having a simple description of a mathematical structure
which determines it up to isomorphism is desirable in many parts of mathematics.
If the description is intended to be in first order logic and the structure
is infinite, this is not possible. There are some structures where their
first order theory together with their cardinality determines their isomorphism
type--these structures are called categorical. I will give a survey of
the subject of categorical first order theories and give some indications
of its connections with combinatorial geometry, algebraic geometry and
permutation groups. No previous knowledge of logic or model theory will
be assumed.
ABSTRACT: In this talk, we construct a class of subalgebras of
the Hall algebra that has the generic property following Lusztig's geometric
construction and then show that (with certain integral form), these subalgebras
can obtained from a positive part of quantum enveloping algebra of generalised
Kac-Moody Lie algebra. This algebras has a Lusztig type canonical basis
with certain interesting irreducible representations. When $Q$ is a cycle,
one can obtain the entire Hall algebra ${\Cal H}(Q,q)$. Note that in this
case, the Hall polynomials exist.
This is a joint work with N. Jing and J. Xiao.
ABSTRACT: The non-commutative and quasi symmetric functions are
dual Hopf algebras that share many of the same properties and are strongly
related to the space of symmetric functions. The quasi-symmetric functions
are attributed to Gessel in 1983 and the non-commutative symmetric functions
have their origins in the early 90's, and hence are fairly new by comparison
to the space of symmetric functions. Much of the theory that is well understood
for the symmetric functions has yet to be generalized to this pair of algebras
and some interesting questions arise in developing analogs of constructions
that are well known for the symmetric functions.
Much of the recent research in symmetric functions has been on the properties of two remarkable bases and their generalizations, the Hall-Littlewood and Macdonald symmetric functions. These bases depend on a parameter q and by specializing the parameter to various values they interpolate many of the well known bases of the symmetric functions. One reason they are of interest is that algebraic identities involving these functions often encode several well known identities in the space of symmetric functions at the same time. We define a possible analog to the Macdonald and Hall-Littlewood bases in the non-commutative symmetric functions that arises by abstracting a formula for the Hall-Littlewood functions to the level of Hopf algebras and then demonstrate some of the surprising properties held by these functions.
This is joint work with Nantel Bergeron.
Geanina Tudose, York University, will speak on "On the combinatorics
of sl(n)-fusion algebra" at 12:30p.m. in N638 Ross.
ABSTRACT: The fusion algebra also known as the Verlinde algebra plays a central role in the 2 dimensional Wess-Zumino-Witten models of conformal field theory. The study of the multiplicative structure of this algebra has received a lot of attention in the past decade due to the fact that it appears in an increasing number of mathematical contexts such as quantum cohomology, representations of quantum groups and Hecke algebras, knot invariants, vertex operator algebras, and others.
The $sl(n)$-fusion algebra can be viewed as a quotient of the ring of symmetric functions in $n$ variables by the ideal generated by Schur functions $S_\lambda$ indexed by partitions of length at most $n$ such that $\lambda_1-\lambda_n \leq k$ and $S_{1^n}-1$.
From representation theoretic arguments it is known that its structure constants N_{\lambda \mu}^{\nu}, called fusion coefficients, are non-negative integers. We will give a combinatorial description for these numbers for $\mu$ two column and hook partitions and a larger family of partitions obtained via fusion invariants. In addition we present a number of applications for these cases including the proof of the conjecture that the fusion coefficients are increasing with respect to the level.
ABSTRACT: Catalan number classically enumerate Dyck paths. We
investigate the quotient ring $R_n$ of the ring of polynomials $\Q[[x_1,x_2,\ldots,x_n]]$
over the the ideal generated by non-constant quasi-symmetric polynomials.
We show that the dimension of $R_n$ is bounded above by the $n$th Catalan
number. [In fact the equality should hold].
ABSTRACT: Control problems for hyperbolic PDEs have many engineering applications and have been the subject of extensive mathematical and computational studies. We plan to extend the mathematical work on this subject into some new settings with new ideas; to propose a new algorithm for solving the control problems (both distributed control and boundary control), i.e., a shooting method based algorithm (which natually leads to parallelizations); to derive error estimates for the algorithm and to test it computationally in various settings. Finally, we propose, analyze and implement a new approach for solving the exact boundary controllability problems. Our computational results demonstrate that our new algorithms work effectively and they represent legitimate alternative methods to those in the extant literature.
PhD students are expected to attend the talk.
ABSTRACT: N. Bergeron and F. Sottile defined the universal Grassmannian
order on the symmetric group in an attempt to better understand the Littlewood
Richardson coefficients for Schubert polynomials. In this talk, we will
give an alternative characterization of this order in terms of basic permutation
statistics, and use this characterization to establish some properties
of this partial order. In addition, we give a third characterization of
this order that naturally generalizes to define a whole class of partial
orders on the symmetric group.
ABSTRACT: Let $A$ be a finite-dimensional (unitary) associative
algebra over an algebraically closed field. Then the multiplicative semigroup
of $A$ is a linear algebraic monoid, of which the unit group is a linear
algebraic group. Several characterizations of nilpotency of the unit group,
in terms of Lie algebra, idempotent, the normalizer of a maximal torus,
subgroup or subalgebra, are discussed.
ABSTRACT: Let $G$ be a reductive group acting linearly on the
vector space $V$. Let $S=k[V]$ be the regular functions on $V$ and $S^G$
be the invariants in $S$ under the induced action of $G$ on $S$. A representation
$V$ is called cofree if $S$ is a free $S^G$-module.
A classical problem in Invariant Theory is to determine for which groups $G$ and spaces $V$ is $V$ a cofree representation.
In this talk, we discuss a new family of representations which are cofree. In particular, given a finite quiver $Q$, we want to show when the action of $SL(Q,d)$ on $Rep(Q,d)$ gives a coffee representation. We present a class of quivers which have this property.
ABSTRACT: If someone hands you a complex built out of Euclidean
polyhedra, then how could you check that it is non-positively curved? In
this talk we will see that there is a practical algorithm that can decide
whether or not a 3-dimensional metric polyhedral complex is locally CAT(0).
We will also see that in fact there is a (not-so-practical) algorithm for
complexes of ANY dimension. The first procedure is very geometric, and
the second uses techniques from computational algebraic geometry.
This is joint work with Jon McCammond.
ABSTRACT: Some basics on block algebra and its invariants such
as the number of ordinary irreducible characters, the number of irreducible
Brauer characters, defect group and its order will be introduced. Then
I will go on to present some results of Cartan matrix in connection with
block invariant.
ABSTRACT: One of the many places in which lattice animals and
their relatives arise is in the modelling of the physical properties of
polymers. Most work in this area has concentrated on polymers made up of
a single type of building block (or monomer), and many results (both exact
and numerical) are known.
Not all polymers are homogeneous, and many interesting polymers (such as DNA) are made up of two or more types of monomers with different properties. There are far fewer exact results for models of inhomogeneous polymers (or co-polymers) and many basic questions about their behaviour remain unanswered.
I will give a LIGHT introduction to some basic statistical mechanics and phase transitions, before proceeding onto some combinatorial models of polymers, some recent exact solutions, and the application of a little number theory
ABSTRACT: In this talk we will adapt to classic Hopf algebras
the theory of Aguiar on infinitesimal Hopf algebra. This will allow us
to see our work on Pieri operations on poset with a new perspective. We
introduce the concept of the Hopf algebra, HP, of a poset, P. Then defining
Pieri operations on the Poset P will naturally induce (via universal properties)
a Hopf-morphism, K, from HP to the Hopf , Qsym, of quasi-symmetric functions.
If time allow, we will see that the peak algebra of Stembridge is then
the natural Eulerian subalgebra of Qsym.
ABSTRACT: In this talk we are going to study the irreducible
modules for a class of vertex subalgebras of the vertex algebras associated
with non-positive definite even lattices. As an application we will also
study the representations for the abelian extension of the generalized
Witt algebra.
The talk should be fairly accessible. Everyone is welcome.
ABSTRACT: A classification of the pairs $(A,D)$ is obtained,
where $A$ is a commutative associative algebra with an identity element
over an algebraically closed field of characteristic zero and $D$ is a
finite dimensional subspace of locally-finite commuting derivations of
$A$ such that $A$ is $D$-simple. Such pairs $(A,D)$ are the fundamental
ingredients of constructing Lie algebras of generalized Cartan type. From
the such pairs $(A,D)$, some new infinite-dimensional simple Lie algebras
can be constructed, which are in general not finitely graded.
ABSTRACT: In this talk we discuss a method to obtain finite dimensional
representations of the braid group of type B via centralizer algebras of
quantum groups. Because of the relationship of quantum groups to Lie algebras,
we are able to use combinatorics of the theory of weights and their correspondence
to Young diagrams to label the irreducible representations.
Through this method, for example, we obtain specializations of the Hecke algebra of type B, Ariki-Koike algebras (also known as cyclotomic Hecke algebras) and the q-Brauer algebra of type B (also known as the B-BMW algebra). These algebras will be defined in this talk.
One application of using the method, described in this talk, is that it allows us to define a trace on these algebras, called the Markov trace. This trace is important in determining when the algebras in the previous paragraph are semisimple.
Part of this talk is joint work with H. Wenzl.
ABSTRACT: The classification of simple Lie algebras over the
reals and complexes has been known for a long time. But over an arbitrary
field, the question is quite a bit more complicated. (For example, in characteristic
$p$, one should probably look at algebraic groups instead of Lie algebras.)
The classification is still very far from being resolved, and some conjectures
due to Serre and others are equivalent to special cases of it.
A survey talk, accessible to all, revealing the origins of the
theory of profinite groups.
ABSTRACT: We explain a basic conjecture on finite dimensional
modules of quantum affine algebras. Partial resuts will be given, from
the algebraic and the geometric point of view. We will give a particular
attention to the geometric approach via the cohomology of quiver varieties.
ABSTRACT: Let V be a simple vertex operator algebra and G a finite
automorphism group of V. A major problem in orbifold conformal field theory
is to determine the module category of V^G. Here V^G is the G-invariant
sub-vertex operator algebra of V. Let S be a finite set of inequivalent
irreducible V-modules. Then there is a finite dimensional semisimple associative
algebra A_{\lapha}(G,S) such that A_{\alpha}(G,S) and V^G form a dual pair
on the sum of V_modules in S in the sense of Howe. In particular, every
irreducible V-module is completely reducible V^G-module.
ABSTRACT: The most important, as well the most efficient way
to determine the structure of a group is to use the information of its
subgroups and quotient groups. Among the subgroups, people pay more attention
to Sylow subgroups, maximal subgroups and minimal subgroups. This talk
will mention some recent works on maximal subgroups, Sylow subgroups, minimal
subgroups in terms of c-normality and c-supplementary.
ABSTRACT: We study the cohomology of various Lie algebras and
differential graded Lie algebras of vector fields on Riemann surfaces.
Topics include the Kodaira-Spencer differential graded Lie algebra and
its relation to deformation theory, its sub Lie algebra of smooth vector
fields of anti-holomorphic type and a 2 dimensional analogue of the Virasoro
algebra, and finally Krichever-Novikov algebras being Lie algebras of meromorphic
vector fields with poles in a fixed set of points. In conclusion we state
some open problems.
ABSTRACT: While the key points of this talk may be considered
(by many experts) MUCH TOO CLASSICAL -- dating back to Schur & Frobenius
a full century ago, there is lot to be desired about the wider dissemination
of the basic facts on Irreducible Representations (= IrReps) of the two
families of groups (in chc 0). The 1939 classic of H Weyl ‘Classical Groups'
is now unreadable, partly due to his systematic avoidance of his own Character
Formula from 1926; our main focus will be how to deduce all the key results
Ab Initio (using almost nothing more than Schur's 2 Lemmas, one ‘trivial'
& well known and the other his main link between the 2 families of
groups -- sometimes called Schur Duality), as also how to place these in
historical perspective going back to early 19-century in the Theory of
Symmetric Functions.
The question in the title is meant to ask for further examples (in the Future) of the same idea of Schur in more excciting cases (that mighht include Representations of the ‘Monster' for instance).
While today it's the understang of Symmetric Group IrReps that needs to reach wider audience (with spreading interests in the Methodology of Enumerative or Algebraic Combinatorics), the somewhat better known facts in the $GL_n$ IrReps need also to be placed in a ‘Generalized MultiLinear Algebra' perspective that produces the fruitful idea of SHUR FUNCTORS. This talk will contain a simultaneous (somewhat fast paced, yet) elementary fashion. Our methods follow the key 19-century idea of ‘polarization' on one hand, & give rise to a cleaner approach to the ‘Tableaux Theory' (that can be developed in a less ad hic way).
ABSTRACT: The talk includes some recent results on certain complex
reflection groups. This includes the relation between the imprimitive complex
reflection groups $G(m,m,n)$, $G(m,1,n)$ and the (extended) affine Weyl
groups $G_0(x,n)$, $G(x,n)$ of type $\widetilde{A}_{n-1}$, the relation
between the corresponding Hecke algebras. We introduce some new presentations
of these groups and some applications. We also give a characterization
of Coxeter groups among all the irreducible finite reflection groups of
rank greater than one.
ABSTRACT: According to McKay there is a one-to-one correspondence
between finite subgroups of SL(2, C) and affine Dynkin diagram of ADE types.
McKay correspondence has been shown to exist in several different fields,
and recently a new form in vertex representations of toroidal Lie algebras
and affine Lie algebras was given by Frenkel-Jing-Wang. I will discuss
an analogous group-theoretic construction of twisted vertex operators using
certain double covering groups of wreath products.
ABSTRACT: Vertex operator algebras are a new class of algebraic
structures which have recently arisen in mathematics and physics. Monstrous
moonshine is about the connection between the monster simple group and
modular functions. In this talk I will review the history of monstrous
moonshine, including the basic facts about vertex operator algebra and
Borcherds' proof of Conway-Norton's moonshine conjecture. I will also discuss
Norton's generalized moonshine conjecture and its relation with orbitfold
conformal field theory.
ABSTRACT: We discuss two combinatorial formulas for a q-analogue
of the level-restricted tensor product multiplicities in type A, in the
case of several tensor factors all indexed by rectangular partitions. This
q-analogue is defined using affine crystal theory. The combinatorial objects
in the two formulas are certain Littlewood-Richardson tableaux and rigged
configurations. The rigged config formulas allow us to compute some new
explicit formulas for some affine type A branching functions. We also give
a q-analogue of level-rank duality for the above crystal level-restricted
q-analogues and a conjectural generalization for the q-analogues of Lascoux-Leclerc-Thibon.
This is joint work with Anne Schilling.
ABSTRACT: Consider the space of symmetric functions \Lambda and
the operators on this space (elements of Hom(\Lambda, \Lambda)). We introduce
an involution which relates pairs of operators that add rows or columns
to the partitions indexing symmetric functions (vertex operators).
A simple q-analog of this involution changes generating functions for Schur vertex operators to generating functions for the Poincare polynomials known as generalized Kostka polynomials. These generating functions allow us to compute explicit relations among the generalized Kostka polynomials that extend many of the relations that are known for the Kostka-Foulkes polynomials.
Finally, we use this involution to produce a combinatorial rule for the action of the operator that adds a column on the homogeneous symmetric functions when this operator acts on the Schur basis (a sort of dual to the Pieri rule since multiplcation by h_k adds a row).
ABSTRACT: Dale Peterson has defined and studied what he calls
"lambda-minuscule" elements of (symmetrizable Kac-Moody) Weyl groups. These
elements can be encoded by, or even identified with, a certain class of
labeled partially ordered sets. In type A, the posets are Young diagrams.
In total, there are 17 "irreducible" families of these posets, 16 of which
have infinitely many members.
As has become increasing clear in ongoing work of Robert Proctor, there is an amazingly rich combinatorial theory hidden in these posets, generalizing much of the classical combinatorics of Young diagrams. For example, there is an explicit product formula, due to Peterson (refined later by Proctor) for the number of reduced expressions for any lambda-minuscule element. This generalizes the famous hooklength formula of Frame-Robinson-Thrall for counting standard Young tableaux.
In this talk, we will survey the subject matter, including various characterizations, classifications, and applications.
ABSTRACT: For each Coxeter group we can partition up the group
elements in a certain way, and form a formal sum from the elements. These
formal sums form the basis of what is known as a descent algebra. Much
study has gone into these algebras over a field of characteristic 0, but
little has been done over a field of finite characteristic. My research
has attempted to formulate some results in this area, and in this talk
I shall discuss what has been found so far, both for certain Coxeter group
families, such as the symmetric groups, and in general.
ABSTRACT: The cores of extended affine Lie algebras of reduced
types were classified except for type $A_1$. The main part of the classification
of these Lie algebras is to determine their coordinate algebras. It turns
out that in the $A_1$-case the coordinate algebra is a certain $\Bbb{Z}^n$-graded
Jordan algebra, called a Jordan torus, which can be considered as a Jordan
analogue of the algebra of Lauren polynomials in $n$ variables. In my talk
I will start by defining extended affine Lie algebras and the cores. Then
I will show how the Jordan tori appear as the coordinate algebra in the
$A_1$-case. After this, I will explain the examples of Jordan tori, and
how the classification works. It turns out that Jordan tori are strongly
prime, and so Zelmanov's Prime Structure Theorem can be applied. I will
briefly show that there are only three classes of Hermitian type, one class
of Clifford type and a unique exceptional Jordan torus, called the Albert
torus.
ABSTRACT: A commutator type operator in $l$ variables is a "multilinear"
operator $T$ that maps any $l$-tuple of subgroups of an abelian group to
a subgroup. Several typical examples for such operators are obtained from
commutators in Lie-rings. We can define the concept of T-automorphism,
$T$-solvability, $T$-nilpotency, $T$-ideal and $T$-section in a similar
way as for Lie-rings.
My talk will focus on the following theorem: "If a finite $p$-group
$P$ admits an automorphism of order $p^n$ with exactly $p^m$ fixed points,
such that $\varphi^{p^{n-1}}$ has exactly $p^k$ fixed points, then $P$
has a fully-invariant subgroup of $(p,n,m,k)$-bounded index which is nilpotent
of $m$-bounded class.". This is a theorem towards conjecture no. 2 of Medvedev's.
After giving historical background and a brief explanation about what are
these animals called "commutator type operator", I will explain how this
theorem is obtained by applying twice a theorem on commutator type operators.
If the time will permit, I will show the idea of the proof of that
theorem on commutator type operators.
ABSTRACT: An irreducible representation of the extended affine
Lie algebra of type $A_{n-1}$ coordinatized by a quantum torus of $2$ variables
is obtained by using the Fock space for the principal vertex operator realization
of the affine Lie algebra $\widetilde{gl}_n$.
ABSTRACT: The maximal minors of a generic rectangular matrix satisfy
interesting quadratic relations. A reduced Groebner basis for the Pluecker
ideal of these relations is elegantly described in terms of a natural Bruhat
order defined on the set of maximal minors. This shows the maximal minors
generate an algebra with straightening law on this Bruhat order, and these
results are classical.
If we now consider a generic matrix of polynomials in a variable
t, then the maximal minors are themselves polynomials in t. The goal of
this talk is to describe a reduced Groebner basis for the ideal of algebraic
relations among these coefficients. This will show that the coordinate
ring of the quantum Grassmannian (a singular compactification of the space
of rational curves in the Grassmannian) is an algebra with straightening
law on a quantum Bruhat order.
We will begin by reviewing the classical situation described above,
including some interesting geometric consequences of the reduced Groebner
basis for the Pluecker ideal. We will then describe a reduced Groebner
basis for the quantum Pluecker ideal, indicate our method of proof, and
describe some consequences of these relations. This talk represents joint
work with Bernd Sturmfels.
ABSTRACT: Each field can be embedded into its algebraic closure,
each R-module has an injective envelope, and each poset has a MacNeille
completion. Question: are these (and similar) constructions functorial,
such that the embeddings become natural transformations? A short and simple
(but slightly clever) categorical argument shows: never! The question makes
particularly sense in the context of a Quillen model category, but here
the answer is no longer easy.
Stephen Berman, University of Saskatchewan/Univerity of Virgina,
will give a talk entitled "A view of extended affine Lie algebras with
some historical perspective" at 2:30p.m. in N638 Ross.
ABSTRACT: Starting with the finite dimensional simple Lie algebras over the complex numbers, the historical background leading to the discovery of the Kac-Moody Lie algebras will be discussed. The most important of the Kac-Moody algebras are the affine Lie algebras and their classification, structural properties, and their many and varied applications will be indicated. We will see how these developments led to the advent of the extended affine Lie algebras (EALA's) and by using particular examples, indicate how they can be understood and classified. Most of this talk should be accessible to a general mathematical audience including graduate students.
ABSTRACT: A group is said to be co-Hopf if it is not isomorphic
to a proper subgroup of itself.
Z.Sela showed that a torsion-free word-hyperbolic group is co-Hopf if it does not split over cyclic groups. He later used the theory of JSJ decomposition to prove that a non-elementary torsion-free word-hyperbolic group $G$ is co-Hopf if and only if $G$ is freely indecomposable. L.Potiagailo obtained similar results for Kleinian groups.
We show that none of these statements holds for finitely generated subgroups of word-hyperbolic groups, by constructing explicit counter-examples.
ABSTRACT: The $n!$ conjecture, stated by A. Garsia and M. Haiman
asserts that the vector space $M_{\mu}$ spanned by all partial derivatives
of a polynomial $\Delta_{\mu}$ associated to the partition $\mu$ of $n$
has dimension $n!$.
The aim of this work is to propose a new way to prove the $n!$ conjecture for some particular partitions. The goal is to construct a monomial and explicit basis for the space $M_{\mu}$. We succeed completely for hook-shaped partitions, i.e. $\mu=(K+1,1^L)$. We are indeed able to exhibit a basis and to verify that its cardinality is $n!$, that it is linearly independent and that it spans $M_{\mu}$. We deduce from this study an explicit and simple basis for $I_{\mu}$, the annulator ideal of $\Delta_{\mu}$. This method is also successful for giving directly a basis for the homogeneous subspace of $M_{\mu}$ consisting of elements of $0$ $x$-degree.
ABSTRACT: Abstract: Lie algebras graded by a finite reduced root
systems were introduced in 1992 by S. Berman and R.V. Moody. Since that
time the structure of such Lie algebras has been determined up to central
extension in terms of coordinate algebras. We will describe how to determine
the central extensions of a root graded Lie algebra in terms of the homology
of its coordinate algebra.
ABSTRACT: We will describe Pieri operators, and then see how
they can be used to identify subalgebras of the Hopf algebra of quasi-symmetric
functions. We will then illustrate this with examples that involve enriched-P
partitions, flag f-vectors, and symmetric functions, amongst others.
ABSTRACT: Root graded Lie algebras of type $A_2$ allow alternative
algebras (like octonions) as coordinates while root graded Lie algebras
of type $A_1$ allow Jordan algebras as coordinates. In this talk, the Seligman
construction for the type $A_2$ Lie algebra and the Tits-Kantor-Koecher
construction for the type $A_1$ Lie algebra will be introduced. Then we
determine their universal central extensions.
ABSTRACT: Central extensions of root graded Lie algebras can
be characterized as certain homology of their coordinates. In this talk,
I will focus on the simplest case: the elementary matrix Lie algebra $sl_n(R)$,
where $R$ is an associative algebra. It turns out that the universal central
extension of $sl_n(R)$ is the first Connes cyclic homology group of $R$
if $n>2$. Some variations of cyclic homology such as K\"ahler differentials,
Hochschild homology and (skew) dihedral homology will also be introduced.
Everyone is welcome.
Professor M.F. Newman, Australian National University, will speak
on "Another Question of Burnside" at 4:00p.m. in N638 Ross.
ABSTRACT: William Burnside is known for many contributions to the theory of groups; in particualr for his questions about simple groups and periodic groups. This talk will discuss another good question of his about groups with prime-power order.
ABSTRACT: Various well-known questions of the type: "Is every
Burnside group finite?" or "Is every n-Engel group nilpotent-by-finite
exponent?" will be considered in a certain large class C of groups including
all groups considered in traditional textbooks on group theory.)
ABSTRACT: The notion of extended affine Lie algebra (EALA, for
short) is a higher dimensional generalization of affine Kac-Moody algebras
introduced by mathematical physicists.
Toroidal Lie algebras are examples of EALAs.
There are many EALAs whose coordinates allow not only the Laurent polynomial algebra but also quantum tori, Jordan tori and the octonion torus depending on the type of EALAs.
For instance, an EALA of type $A_{n-1}$ can be coordinatized by a quantum torus $\Bbb{C}_q$ associated to a $\nu\times\un$ matrix $q$.
In this talk, I will survey some recent developments on EALAs. Then I will construct an irreducible vertex operator representation for the EALA of type $A_{n-1}$ coordinatized by a quantum torus $\Bbb{C}_q$ of $2$ variables.
Everyone is encouraged to attend.
DR GAO IS A CANDIDATE FOR THE ALGEBRA/NUMBER THEORY POSITION.
ABSTRACT: In 1977 I proved the following generalization of Magnus'
Freiheitssatz: In any presentation of a group via n generators and m relations
(n>m), some n-m of these generators will freely generate a free group.
I shall talk about generalizations of this result to certain factor groups
of free products of appropriate groups.
ABSTRACT: For a finite group $G$, a $\z G$ lattice $A$ and a
field $K$ on which $G$ acts trivially, the action of the group $G$ can
be extended naturally to an action on the quotient field $K(A)$ of the
group algebra $K[A]$. The fixed field $K(A)^G$ of $K(A)$ under this action
of $G$ is called a multiplicative invariant field. We prove that the rationality
problem for a given multiplicative invariant field $K(A)^G$ over $K(A^R)^G$
where $R$ is the normal subgroup of $G$ generated by reflections acting
on the lattice $A$ is equivalent to the rationality problem for the multiplicative
invariant field $K(A)^{\Omega_G}$ over $K(A^R)^{\Omega_G}$, where $\Omega_G$
is a particular subgroup of $G$ satisfying $G/R\cong \Omega_G$. We use
this result to prove that $K(A)^G$ is rational over $K$ where $G$ is the
automorphism group of a crystallographic root system $\Psi$ acting on $V=
\q\Psi$ and $A$ is any $\z G$ lattice on $V$.
ABSTRACT: What is the free field? The talk is intended to answer
this question, and to show some links with automata theory and linear recursive
sequences. Roughly speaking, the free field is a noncommutative analogue
of the field of rational functions. It is not easy to construct, since
the construction with fractions does not work. One possibility is to embed
it into the ring of Malcev-Neumann series on the free group (Lewin). Another
one is to develop, as Paul Cohn does, a noncommutative theory of localization.
Another one is to view it as the set of rational expressions modulo the
universal identities (Amitsur).
ABSTRACT: An $n-$ary word $w$ is called an $n-$symmetric word
in a group $G$ (or simply: a symmetric word) if (*) $w(g_1, \ldots, g_n)=w(g_{\sigma(1)},
\ldots, g_{\sigma(n)})$ for all $g_1, \ldots, g_n$ from $G$ and all permutations
$\sigma$ from the symmetric group $G$.
Symmetric words in a given group $G$ are closely connected with fixed points of the automorphisms permuting generators in the corresponding relatively free group, and also with symmetric operations in universal algebras and symmetric identities.
The $n-$symmetric words in $G$ form a group $S^{(n)}(G)$.
I give a survey of results concerning generators of $S^{(n)}(G)$ in nilpotent, soluble and finite groups.
ABSTRACT: In 1992 Wilson and Zelmanov proved that a profinite
Engel group is locally nilpotent. Here we prove the stronger result that
every compact Engel group is locally nilpotent. In the talk we will discuss
Wklson and Zelmanov's approach and give ideas of a new proof. As usual
we will talk about open problems.
ABSTRACT: Even though they have been studied for over a hundred
years, the Rogers-Ramanujan identities are still of great interest today
(there is even a connection to the 1998 Nobel prize in physics!). I will
survey some of the recent developments related to the study of Rogers-Ramanujan-type
identities and also discuss problems still to be solved.
ABSTRACT: The origins of Iwasawa theory can be traced back to
the famous class number formula of Dirichlet, which states that the number
of inequivalent binary quadratic forms of a given fundamental discriminant
$D$ is determined by the value $L(1,\chi_D)$ of the associated quadratic
L-function. In modern terms, the class number formula is a very special
case of a {\em Main Conjecture in Iwasawa theory}, which relates analytic
information (as given by L-values) to algebraic information given by cohomology
(the so-called Selmer group, which reduces in the classical case to quadratic
forms and genus theory). The goal of this talk is to give an introduction
to the theory of $p$-adic L-functions, and to describe some recent results
in the subject.
A more detailed description is as follows. We will introduce the classical theory of $p$-adic L-functions attached to cyclotomic fields. We will define the L-functions and Selmer groups, and show how classical questions on the zeroes and poles of complex L-functions have precise $p$-adic analogues. We will state the main conjecture, which provides a precise link between the algebra and the analysis. We illustrate the theory by describing the proof of the $p$-adic Artin conjecture. In conclusion, we introduce some new results in the Iwasawa theory of modular forms and elliptic curves.
ABSTRACT: I will define Piere operations on POSET and show that
they unify many construction of some special functions. In particular,
Skew-Schur functions, Skew-Schubert symmetric functions, Stanley F symmetric
functions, Ehrenborg quasi-symmetric functions, ... I will give many examples
and open problems.
Aner Shalev, Hebrew University of Jerusalem, will speak on "Group
and Probability" at 3:00p.m. in N638 Ross.
ABSTRACT: The Fields medal is math's most prestigious honour,
and in august Richard Borcherds recieved one. It turns out that my work
overlaps somewhat with his (sigh, his work is much deeper....), and so
I can perhaps explain a little of what he has accomplished. The main thrust
of his work was to understand why 196884=196883+1, and to accomplish this
he invented vertex operator algebras and generalised Kac-Moody algebras.
ABSTRACT: A subgroup H of a group G is supplemented in G if there
is a subgroup K of G such that G=HK. We investigate groups in which all
subgroups, all normal subgroups, or all characteristic subgroups have a
proper supplement. This supplement can be either an arbitrary subgroup,
a normal or a characteristic subgroup, resulting in nine classes of groups.
Properties of these classes are studied such as containment and closure
properties, and characterizations for several of these classes are given.
ABSTRACT: Compact torsion groups have been studied in the context
of harmonic analysis, but, being profinite groups, their study leads to
questions connected to the restricted Burnside Problem. During the last
ten years enormous progress has been made (E. Zelmanov, Y. Medvedev). In
my talk I shall present some of the earlier results due to J. McMullen
Herfort, J. Wilson and try comment on it.
ABSTRACT: A generalized {shape} such as
1 2 3 4 5 6 1 x x x 2 x x x x 3 x 4 xis a collection of cells in a matrix. Just as {partition} shapes such as
1 2 3 4 5 6 1 x x x x x x 2 x x x x 3 x x x 4 x xare assoicated to representations of $S_n$ and $GL_n$, so is each generalized shape.
If $D$ is a shape, the associated representation ${\cal S}^D$ may be constructed as the linear span inside the polynomial ring ${\bf C} [x_{i,j}]$ of certain products of determinants of minors in the matrix $(x_{i,j})$. Having made this construction, one can order the monomials of ${\bf C}[x_{i,j}]$ and ask to
In this talk, I describe how (under suitable monomial orders)
to fully answer the above questions when $D$ is a "row-convex" shape, or
even when $D$ is replaced with a union of "row-convex" shapes. I conjecture
two structural results concerning the answer to problem~(1) and I describe
how to verify these conjectures on a different class of shapes using some
new techniques of Bruns and Conca.
Applications include "{\sc sagbi}-basis" algorithms for the coordinate rings of some configuration varieties and (via a result of Sturmfels) some easy proofs that some of these rings are Cohen-Macaulay.
ABSTRACT: We will present the "classical" theory of harmonics
of the symmetric group (as a special case of results of Steinberg and others),
and generalizations suggested by the study of representations theoretic
models for Macdonald symmetric polynomials (although no knowledge of these
polynomials will be used). The only background needed will be basic linear
algebra and algebra, and some combinatorics, and even with this restriction
we will be able to present exciting new (and not so new) conjectures. If
time allows, we will suggest how deeper conjectures can be obtained in
the framework of representations of the symmetric group.
ABSTRACT: Every compact periodic group is totally disconnected.
This is not true for compact Engel groups. In my first talk in November,97
we discussed several conjectures on a structure of compact Engel and strongly
Engel groups which are based on corresponding results for profinite groups
due to Wilson, Zelmanov, Medvedev. I will discuss a distance from profinite
groups to arbitrary compact group and talk about a solution of one of these
conjectures. Namely, I will show that a strongly Engel compact group is
totally disconnected modulo of a closed nilpotent subgroup. Actually, any
such group is just an extension of a nilpotent group by a group of finite
exponent.
ABSTRACT: Many naturally arising sets of permutations are closed
under the operation of taking subsequences. Some examples will be given
and the main questions introduced with partial answers being given in special
cases. The talk requires no specialized knowledge and will be intelligible
to graduate students and faculty in Pure Mathematics.
ABSTRACT: In 1971 Ryogo Hirota invented a method to produce exact
N-soliton solutions for many important non-linear partial differential
equations. Ten years later it was discovered by the Kyoto school that there
is a hidden Kac-Moody group action on the space of solitons in the Hirota's
method. Another important ingredient of this theory is the construction
of the vertex operators discovered by physicists in the string theory.
One obtains soliton solutions of the hierarchy of non-linear PDEs from
the representation of Kac-Moody algebras by vertex operators.
ABSTRACT: Algebraic combinatorics is interested in things like
tensor product coefficients for Lie algebra representations. A "truncated"
or "folded" version of these appears naturally in places like quantum groups,
modular representations of Chevalley groups, or conformal field theory,
and is given by the celebrated Verlinde formula.
This talk is a TECHNICAL INTRODUCTION to a number of new and relatively unexplored questions, which really should belong to algebraic combinatorics.
I will assume some knowledge of complex Lie algebra representations.
ABSTRACT: Consider ordered factorizations of an arbitrary element
of the symmetric group on n symbols, into factors which all belong to the
same conjugacy class. An expression for the number of such factorizations
is available via characters, but in certain cases a more compact answer
can be obtained using a family of symmetric functions constructed by Macdonald.
With the further restriction that the group generated by the factors acts
transitively on the n symbols, we call these transitive factorizations.
Hurwitz considered the number of transitive factorizations in the case
that the factors are transpositions, because of its connection with counting
all nonequivalent ramified coverings of a Riemann surface. In this case
he gave a remarkably compact, explicit answer. In this talk, extensions
of Hurwitz' result are presented, suggesting that transitive factorisations
have an elegant structure, including a close, but as yet unknown, link
with Macdonald's symmetric functions.
ABSTRACT: We shall recall some details about symmetric functions,
and define a certain set of minimal idempotents in the descent algebra
of the symmetric group. From here we shall set up the problem I am working
on at present, which concerns finding a new basis for these idempotents
whose coefficients lie inthe ring of symmetric functions. We shall then
motivate why we should want to look for such a basis.
ABSTRACT: I will define some new combinatorial objects related
to root systems and make a list of open problems in this theory. These
new objects are generalizations of standard Young tableaux and their combinatorics
gives information about the representation theory of affine Hecke algebras.
Any solutions to the problems that I will pose would have interesting representation
theoretic implications. There are many more problems than I have time to
work on and I would be thrilled to have others take some of them up!
The basic preliminary concepts for Arun Ram's talk are:
1) Irreducible representations are indexed by shapes
2) Skew shape representations also make sense
3) Bases of the irreducible representations and skew shape representations are indexed by standard Young tableaux
4) The Littlewood-Richardson rule tells how a skew shape representation decomposes
5) Finite Coxeter groups are generalizations of symmetric groups and are controlled by hyperplane arrangements
6) Nobody knows much about the combinatorics of irreducible representations of finite Coxeter groups (except in types A,B,D, which are essentially all done by doing type A and then fiddling with it to get the others)
Also, we should point out that for us irreducible representation means simple module. We hardly ever work with matrices when we talk about this stuff, always vector spaces with actions.
ABSTRACT: We first explain how the actions of Hopf algebras naturally
arise when people consider graded algebras, automorphism groups and derivation
algebras. Then we discuss the following problem. Let A be an algebra with
an action of a finte-dimensional Hopf algebra H, and AH the subalgebra
of invariants of the action. Suppose AH satisfies a non-trivial identity
(say, commutative or nilpotent). For what H can one conclude that also
A satisfies a non-trivial identity? Among several results to be mentioned
we formulate the following solution of a problem due to A. Zalesskii. Let
L be a Lie algebra graded by a finite group G and L1 satisfies a non-trivial
identity of degree d. Then L satisfies such an identity of degree f(d,|G|)
depending only on d and |G| but not on L itself.
ABSTRACT: We define skew Schubert polynomials using the skew
divided difference operators introduced by I.Macdonald. A connection with
the structural constants for Schubert polynomials, and other properties
of the skew Schubert polynomials will be discussed.
ABSTRACT: While studying multiplication of Schubert polynomials
with Frank Sottile, we have discover an interesting partial order for the
symmetric group. We will present a monoid associated with this partial
order and discuss some open problems related to this.
ABSTRACT: We define the descent algebra of the symmetric group
over the rationals. We then extend this definition to a field of finite
characteristic. From here we discuss why these structures are of interest,
and formulate one of the many open probelms in this area.
ABSTRACT: The question whether a compact periodic group has a
finite exponent has been known since early 60's. One can find a discussion
on this question in the book on Abstract Harmonic Analysis by Hewitt and
Ross. Recently, Zelmanov, developing his ideas on the solution of the restricted
Burnside problem, has proved that compact periodic groups are locally finite.
Hence, every Sylow p-subgroup of compact periodic groups are Engel. I will
give an outline of an approach which leads to a necessary and sufficient
condition for a periodic compact group to be of finite exponent. It turned
out that a periodic compact group has a finite exponent if and only if
it is strongly Engel. As usual, we will talk about open problems in this
area.
ABSTRACT: Professor Gannon's talk will be about certain finite
sets of complex numbers which are remarkable for two reasons:
(i) There is a long list of diverse contexts in which they arise, such as elements of finite order in Lie groups, quantum gorups at roots of 1, moduli spaces of semistable bundles over Riemann surfaces, Chevalley groups for Z/pZ, representation theory of Kac-Moody algebras;
(ii) They inherit fromthese algebraic contexts many symmetries and properties which makes the theorems and the proofs very pretty.
The theory exploring these sets is still poorly developed, and there are many easy-to-understand open problems. T. Gannon will develop at least one of these contexts, describe some of the known results, and state some of the open problems.
ABSTRACT: Professor Bergeron will give an introduction talk on
Schubert polynomials. In particular he will try to answer the following
questions: What are the Schubert polynomials? Why do we care? What do we
know? Are there some open problems? And then try to answer the questions
from the audience.
[This is related to joing work with F. Sottile.]
ABSTRACT: Let C_k be the cyclic group {1,q,...,q^{k-1}} and Z[C_k]
the algebra of polynomials in the variables 1, q, q^2, ..., q^{k-1} with
coefficients in Z. We introduce Weighted Rooted Necklaces, which give a
combinatorial interpretation of Z[C_k]. Then, an invertible, non-linear
operator R:N[C_k] --> N[C_k] ("N" is the set of non-negative integers)
is also introduced: R(f[q]) = q^{-f[0]} ( f[q]-f[0]+\sum_{i=1}^{f[0]}q^i
). The operator R creates equivalence classes on N[C_k]. We determine a
recursive function to count the equivalence class containing G[q] = \sum_{i=0}^k
q^i in Z[C_k]. We give a closed form for such recursive formula in some
instances. We also develop a series of related questions. The Algebraic
Combinatorics setting we establish here was motivated in the first place
by a "hands on" combinatorics problem first posed by Cipra in 1992 (such
problem is still open in full generality.)