MAAGC 2015 ABSTRACTS:Angela Gibney (U. Georgia): Vector bundles of conformal blocks on the moduli space of curvesI will talk about general aspects of vector bundles of conformal blocks on the moduli space of curves, focusing on recent results and basic open problems. Mark Shimozono (Virginia Tech): Elliptic Hall algebra via symmetric function operatorsI will give an expository talk on the elliptic Hall algebra / shuffle algebra / spherical DAHA emphasizing its realization as an algebra of operators on symmetric functions.
We begin by recalling a beautiful result of Solomon concerning exterior powers of the reflection representation of a Weyl group W and then explain a conjectural generalization of it to Springer representations. The conjecture is known in some cases (Henderson), but a weak version of it (conjecture of Lehrer-Shoji) is always true and this leads to a decomposition of a (singly-graded) generalized Parking Function Module in terms of Springer representations. The decomposition yields polynomials in q (for certain integral parameters m) attached to each nilpotent orbit of the corresponding Lie algebra and each local system on the orbit. When the orbit takes a certain form, these polynomials are related to the characteristic polynomials of a hyperplane arrangement attached to the orbit and they turn out to have non-negative coefficients. The polynomials are also q-analogues for numbers that show up in the combinatorics of the non-crossing partition lattice of W (in type A they yield q-analogues of the Catalan numbers, the Narayana numbers, and the Kreweras numbers). This leads to a conjecture describing the orbit structure of a natural cyclic group action on subsets of the non-crossing partition lattice (known as cyclic sieving), which is true in classical types. This is joint work with Vic Reiner.
The geometry of symmetric orbit closures in the flag variety arises in the representation theory of the real forms of complex semisimple (reductive) Lie groups. In the case of the symmetric pairs (GL(n,C),K), Benjamin Wyser and the speaker have described analogues of the Schubert polynomials. I'll explain this work, as well as an ongoing project with Alexander Woo and Benjamin Wyser to understand the singularities of the orbit closures via combinatorics. |