Tufts Algebra, Geometry, and Number Theory Seminar
2023-04-13
Speaker: Elijah Bodish (MIT)
Title: Fundamental tilting modules for the quantum symplectic group
Abstract: For a simple Lie group the irreducible representations with fundamental highest weight generate the category of all finite dimensional representations. In the case of classical groups, one hopes to find constructions of the fundamental modules using linear algebra. For SL(V), the fundamental representations are exactly the non-trivial exterior powers of V. For the symplectic group Sp(V), the fundamental representations are kernels of the operators “contraction with the symplectic form” acting on the exterior powers of V.
The contraction operator and the ``multiplication by the symplectic form” operator generate an action of SL(2) on the exterior algebra of V. The SL(2) action commutes with Sp(V), and each action generates the others centralizer. Such phenomenon is referred to as Howe duality.
A major open problem in modular representation theory is to compute the characters of tilting modules. The slightly more well-known problem of determining characters of irreducible modules for symmetric groups in finite characteristic is a subproblem. Adamovich and Rybnikov studied some positive characteristic versions of Howe duality. McNinch then leveraged these dualities to, among other things, decompose the exterior powers into indecomposable tilting modules and then describe the characters of fundamental tilting modules for Sp(V).
In my talk I will discuss recent work with Daniel Tubbenhauer (https://arxiv.org/abs/2303.04264) which generalizes the Sp (V)-SL(2) duality to the case of quantum groups. I will also mention a non-trivial canonical basis for the exterior algebra which realizes the Weyl module filtration, and how to compute the characters of fundamental tilting modules for quantum groups.