Abstract:

 The talk aims to introduce two problems I am thinking about. I will first give a new proof (joint with Arthur L. B. Yang) of the (generalized) Jacobi-Trudi identity via the BGG category O of sl_n(C). Then the talk will be devoted to the Stanley-Stembridge conjecture about the chromatic symmetric function, which can be reformulated using the immanants of the Jacobi-Krudi matrices. Finally, I will talk about Haiman's conjecture on the evaluation of virtual characters of Hecke algebra of the symmetric group on the Kazhdan-Lusztig basis, which implies the Stanley-Stembridge conjecture.