On the centers and cocenters of the cyclotomic Hecke and KLR algebras

In this talk, I shall report our work on the centers and cocenters of the cyclotomic Hecke and KLR algebras. We characterize minimal length elements in each conjugacy class of the complex reflection group G(r,1,n) and construct integral bases for the cocenter of the corresponding cyclotomic Hecke algebra. We propose a general approach to study the center conjecture for the cyclotomic KLR algebras. The talk are based on some joint work with Shi Lei, Lin Huang, and with Shi Xiaolin and Shi Lei