Seminar by Anna Lachowska (EPFL)

Title: Small quantum groups and diagonal coinvariants in type A

2016.11.18 | Jane Jamshidi

Date Fri 09 Dec
Time 15:15 16:15
Location 1531-215 (Aud-D3)

Abstract

The question of the structure and dimension of the center of the small quantum group u_q at a root of unity has been open since the object was defined by Lusztig in 1990. A lower bound on the dimension of the center was found in the work of Brown and Gordon, following a similar result for the positive characteristic by Mirkovich and Rumynin. In our work with Bezrukavnikov, the structure of the center of the principal block of u_q was described in terms of certain sheaf cohomologies over the Springer resolution. However, this description did not provide an immediate answer for the dimension of the center for rank higher than 1.

Based on the geometric description, we develop a simple method to compute the dimension of the center of the principal block of u_q corresponding to a semisimple complex Lie algebra g, and present the answers in cases g=sl_3 and g=sl_4. This allows us to formulate a conjecture relating the center of the principal block with Haiman’s diagonal coinvariant algebra of the symmetric group S_n. Applying the same method to the singular blocks of u_q leads to an intriguing combinatorial conjecture for the dimension of the whole center of u_q in type A.

This is a joint work with Qi You (Yale University). 

Seminar