Thursday September 29, 2011, 1:15 PM
Sergi Elizalde (Dartmouth)
A statistic-preserving involution on lattice paths
Abstract: Let T and B be two lattice paths with steps N=(0,1) and E=(0,1), from (0,0) to (a,b), with B not going above T. Let S(T,B) be the set of lattice paths with steps N and E from (0,0) to (a,b) that lie between T and B. For each path P in S(T,B), let
t(P) = number of E steps where P and T coincide,
b(P) = number of E steps where P and B coincide.
Thursday October 13, 2011, 1:15 PM
Dana Ernst (Plymouth State University)
Diagram algebras as combinatorial tools for exploring Kazhdan--Lusztig theory
Abstract: Kazhdan--Lusztig polynomials are subtle objects that arise naturally in the context of Hecke algebras associated to Coxeter groups. Unfortunately, computing these polynomials efficiently quickly becomes difficult, even in finite groups of moderate size. Computing the Kazhdan--Lusztig polynomials would be simplified if one could easily obtain the leading coefficients. In this talk, I will discuss how methods from the theory of diagram algebras can be used to combinatorially and non-recursively compute the leading coefficients of certain Kazhdan--Lusztig polynomials. In particular, we will focus our attention on Hecke algebras of types $A$, $B$, and affine $C$. Moreover, we will relay the current state of affairs of diagram algebras as combinatorial tools for exploring Kazhdan--Lusztig theory.
Thursday October 20, 2011, 1:15 PM
Elizabeth Beazley (Williams College)
A Graph Encoding a Partial Ordering on the Affine Symmetric Group
Abstract:We will introduce several combinatorial models for the affine symmetric group, which is an infinite analog of the group of permutations. We then define an important partial ordering on this group, called the Bruhat order. Amazingly, the Bruhat order on this infinite group is encoded in a user-friendly format by paths in the quantum Bruhat graph, which is a weighted, directed graph whose vertices are the elements in the finite symmetric group. Time permitting, we will discuss applications to quantum Schubert calculus and related open problems.
Thursday November 3, 2011, 1:15 PM
Nan Li (MIT)
ombinatorial aspects of the hypersimplex
Abstract: Given a polytope, we can define its h-vector, where each term is nonnegative and their sum equals the normalized volume of the polytope. the normalized volume ofIt is well-known that the hypersimplex is the Eulerian number. Therefore its h-vector provides a refinement of the Eulerian number. We proved a conjecture by Stanley describing this refinement by descents and excedances by a shellable triangulation. We also generalized this result to slices of larger rectangles. As a byproduct, we came up with a new Eulerian statistic, which enjoys some nice equal joint-distributions with some other known permutation statistics. Combinatorial proofs of some of them are still open.
Thursday November 10, 2011, 1:15 PM
Peter Tingley (MIT)
Affine Mirkovic-Vilonen polytopes
Abstract: Kashiwara developed combinatorial objects called crystals to study the representation theory of complex simple Lie groups and Lie algebras. The construction is quite involved, but one can often realize the same combinatorics by more elementary means. One useful realization is based on the Mirkovic-Vilonen polytopes of the title. I will describe what these polytopes are, and why they are interesting. I will then explain current work giving an analogous construction for symmetric affine Kac-Moody algebras. For affine sl(2) the construction is purely combinatorial. For other symmetric affine types the definitions are combinatorial, but we need some geometry (quiver varieties) to prove that everything works. I will explain these ideas mainly via examples and pictures, and will not assume familiarity with the representation theory involved. This is joint work with Pierre Baumann, Thomas Dunlap and Joel Kamnitzer.
Thursday November 17, 2011, 1:15 PM
Zajj Daugherty (Dartmouth)
Type C symmetry in type A representation theory
Abstract: We'll see how certain kinds of tensor spaces for Lie Algebras g=gl_n or sl_n carry type C symmetry, and how we can use that fact to build an action of the type C Weyl group on modules for the centralizer of g on tensor space of a specific form.