RT \cup QA is a weekly seminar in representation theory and quantum algebra.
It is currently on hiatus for the Winter 2013 quarter due to an absence of key personnel. Talks may be both research and expository. Please do contact the organisers and express your keen desire to speak. Same for adding yourself to our mailing list.

Contact: Peter McNamara

Date

Speaker

Title

October 8

Carl Mautner (Harvard)

A symmetry in the sheaf theory of nilpotent cones
Associated to any reductive group G is an interesting singular variety - the nilpotent cone. Springer, Lusztig and others have shown that by studying the geometry of this variety, one can obtain lots of information about the representation theory of the Weyl group and finite groups of Lie type. In this talk, I will begin by briefly sketching some of the structure of these varieties and its connection to representation theory. Then I will explain joint work with Pramod Achar, in which we show the existence of a symmetry in this structure. More precisely, we find an autoequivalence of its equivariant derived category of sheaves with coefficients in arbitrary fields. As a corollary, we find a good description of the endomorphism rings of many projective objects in the category of perverse sheaves.

October 15

Peter McNamara

A cluster algebra
We will discuss a cluster algebra arising from the theory of algebraic groups, and try to understand what sort of categorical structure might induce the cluster algebra structure. This may involve preprojective algebras, quantum affine algebras or KLR algebras.

Quantum Groups and the Jones Polynomial Given a special kind of Hopf algebra, one can obtain knot invariants. We will do this by constructing a functor from the category of ribbon tangles to the category of representations of a ribbon Hopf algebra. If we forget about the width of a ribbon tangle and work with knots, they will be sent to endomorphisms of the trivial representation which are rational functions. Using this method, we obtain the Jones polynomial if we start with the Hopf algebra U_q (sl_2).

November 12

No

Seminar

November 19

Thanksgiving Recess

November 26

Eric Marberg (MIT)

Positivity conjectures from Kazhdan-Lusztig theory on twisted
involutions Let (W,S) be a Coxeter system with an involution * which
preserves the set of simple generators S. Lusztig has shown recently that
the corresponding set of twisted involutions (i.e., elements of W with
w^{-1} = w^*) carries an essentially unique structure as a module of the
Hecke algebra of W. There is an interesting sense in which this module
serves as an analogue of the regular representation of the Hecke algebra on
itself, and Lusztigâ€™s work indicates that we may repeat much of the
structure theory for the regular representation in the context of the
twisted involution module. For example, the regular representation of the
Hecke algebra of W has a special basis whose coefficients define the
much-studied family of Kahzdan-Lusztig polynomials. In the context of
twisted involutions there is also a distinguished "Kazhdan-Lusztig basis"
whose coefficients define a family of "twisted" Kazhdan-Lusztig
polynomials. In this talk I will give a brief tour of Kazhdan-Lusztig
theory, then describe how this theory emerges as a special case of
constructions attached to twisted involutions, and finally discuss recent
progress on some positivity conjectures associated with the twisted
Kazhdan-Lusztig polynomials.