Abstract
This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $(\mathbb{C}^n)^{\otimes k}$, where $\mathbb{C}^n$ is the reflection representation 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 establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $(\mathbb{C}^n)^{\otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras. We conclude the paper by giving Jucys-Murphy elements of Tanabe algebras and their actions on the Gelfand-Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.
Highlights
The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras
The symmetric group Sk acts on the k- fold tensor product V ⊗k of the n-dimensional vector space V = Cn over the field of complex numbers C
The general linear group GLn(C) acts on V ⊗k diagonally where V is the defining representation of GLn(C). These two actions commute; they generate the centralizers of each other. This is known as the classical Schur–Weyl duality [6]
Summary
Orellana [19] defined a subalgebra Tk(n, r) of partition algebra CAk(n), and proved Schur–Weyl duality between Tk(n, r) and G(r, 1, n) [19, Theorem 5.4]. She recursively constructed the Bratteli diagram for the tower of algebras. We define a subalgebra, denoted by Tk(r, p, n), of partition algebra CAk(n) such that there is Schur–Weyl duality between Tk(r, p, n) and the complex reflection group G(r, p, n). And (ii) we index the components in a w-tuple from 1, . . . , w, for a multiple t of w, the t(mod w)-th component means the w-th component
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
