题目：Interplays between symmetric groups and general linear groups : from classical to quantum, with a view towards categorification

报告人：Marc Rosso，巴黎西岱大学（Université Paris Cité）

时间：2026年10月21日（星期三）下午4: 00-5: 00

地点：第五教学楼5106教室

报告摘要：

The symmetric group S(n) on n variables is a subgroup of the general linear group Gl(n) which leads to the classical Bruhat decomposition. Another connection between S(k)  or Gl(n)  (for different values of k and n) is via representations , in the classical Schur-Weyl duality in the decomposition of tensor powers of the fundamental representation of Gl(n) ( bicommutant theorem). Lying behind is the Hopf algebra structure of the group algebra.

Characters of Gl(n) are symmetric functions, and more generally one may consider the algebra of symmetric functions in infinitely many variables. It has a nice Hopf algebra structure, which can be recovered from the chain of representation groups of symmetric groups using induction and restriction. This is a first example of categorification.

From the Bruhat decomposition of GL(n) on a finite field, one constructs a natural convolution algebra : the Hecke algebra, (a sort of deformation of the group algebra of the symmetric group). It has a nice action on tensor powers of a given vector space V and the quantum group associated with GL(V) has a rich Hopf algebra structure which provides again a bicommutant type situation (quantum Schur-Weyl duality).

All the terms will be recalled or defined, and some further applications and generalizations will be proposed, if time permits.