Alg/Comb/Geom seminar
Fall 2016

There is a geometric version of the Fourier transform that we meet in analysis. We will study this geometric Fourier transform for certain sheaves on quiver representation varieties. I will spend some time talking about the basics of quiver representations, compute some examples of the Fourier transform there, and explain how we can do it all combinatorially for quivers which are linearly ordered type A Dynkin diagrams. This is joint work (in progress) with Pramod Achar and Maitreyee Kulkarni.

The affine Hecke algebra is related to points configurations on the punctured plane as well as representations of p-adic groups; the double affine Hecke algebra is related to points configurations on an elliptic curve as well as representations of p-adic loop groups. In this talk I will introduce a missing link between these two algebras, called the elliptic affine Hecke algebra. It has a construction using equivariant elliptic cohomology. Its irreducible representations are parameterized by nilpotent Higgs bundles on an elliptic curve. This talk is based on my joint work with Changlong Zhong.

We will introduce the combinatorial theory of orthogonal polynomials developed by Viennot (1985) and Flajolet (1980) and then see how it can be applied to the enumeration of skew Young tableaux with at most three rows. In particular, a sequence is the moment sequence for some orthogonal polynomials if and only if it enumerates Motzkin paths with certain weights. In the simple unweighted case, the polynomials are a variant of the Chebyshev polynomials. We will use insights from the combinatorial theory of orthogonal polynomials to give a combinatorial enumeration of skew Young tableaux with at most three rows and fixed skew part as a linear combination of Motzkin numbers. We obtain a simple combinatorial proof of a refinement of earlier results. This follows work by Zeilberger (2006), Regev (2009), Sen-Peng Eu (2010) and Jong Hyun Kim (2011).

Abstract TBA

Given a permutation (or more generally an element in a finite reflection group) w, one can define a hyperplane arrangement called the inversion arrangement. On the symmetric group (or any finite reflection group), one can define a partial order known as Bruhat order. Hultman showed that the number of chambers of the inversion arrangement is always at most the number of elements less than or equal to w in Bruhat order, and gave a condition on the Bruhat graph (a graph related to Bruhat order) for when equality occurs.

This result of Hultman generalizes work of Hultman, Linusson, Shareshian, and Sjostrand in the case of permutations. In this case, they show equality occurs precisely when w pattern avoids the 4 permutations 4231, 35142, 42513, and 351624. This set of permutations was earlier studied in a different context by Gasharov and Reiner, who characterized these permutations as those whose lower Bruhat intervals are defined by inclusion relations. I will talk about a potential generalization of the Gasharov-Reiner conditions defining this set, which I can prove is equivalent to the Hultman condition for type B (which is the hyperoctahedral group, the symmetry group of the n-cube). The type B elements can also be characterized by a list of 31 pattern avoidance conditions.

Abstract TBA