2.00 pm, Seminar Hall
On representation theory of partition algebras for complex reflection groups
Ashish Mishra
Visiting professor, Universidade Federal do Para.
17-01-19

---

Abstract

In this talk, we define the partition algebra, denoted by T_{k}(r,p,n), for complex reflection group G(r,p,n) acting on the k-fold tensor product of the reflection representation C^n of G(r,p,n). A basis of the centralizer algebra of this action of G(r,p,n) was given by Tanabe and for p=1, the corresponding partition algebra was studied by Orellana. We also define a subalgebra T_{k+1/2}(r,p,n) and establish this subalgebra as partition algebra of a subgroup of G(r,p,n) acting on the k-fold tensor product of C^n. We call the algebras T_{k}(r,p,n) and T_{k+1/2}(r,p,n) as Tanabe algebras. Our aim is to study the representation theory of Tanabe algebras: parametrization of their irreducible modules by studying the decomposition of the k-fold tensor product of C^n as a G(r,p,n)-module, and construction of Bratteli diagram for the tower of Tanabe algebras.

We also describe Jucys-Murphy elements of Tanabe algebras and their actions on the Gelfand-Tsetlin basis, determined by the multiplicity-free tower, of irreducible modules.