Algebra and Geometry Seminar

The question of the dimension of the center of the small quantum group u_q(g) at a root of unity has been open since the object was defined by Lusztig in 1990. Until recently the answer was known only for g=sl_2. Based on the description of the blocks of the small quantum group via an equivalence with a category of sheaves over the Springer resolution, originated in the work of Arkhipov, Bezrukavnikov, and Ginzburg, we develop an algorithmic method for calculating the dimension of the regular and singular blocks of the center. The answers obtained in cases sl_3 and sl_4 allow us to formulate a conjecture relating the center of the principal block of u_q(sl_n) with Haiman's diagonal coinvariant algebra of the symmetric group S_n. Applying the same method to the singular blocks leads to an intriguing combinatorial conjecture for the dimension of the whole center of u_q(g) in type A. This is a joint work with Qi You (Caltech).