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.