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Kyungpook Mathematical Journal 2022; 62(1): 1-28

Published online March 31, 2022 https://doi.org/10.5666/KMJ.2022.62.1.1

Copyright © Kyungpook Mathematical Journal.

On the Decomposition of Cyclic G-Brauer's Centralizer Algebras

Annamalai Vidhya and Annamalai Tamilselvi∗

Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai - 600 005, India
e-mail : vidhyamath@gmail.com and tamilselvi.riasm@gmail.com

Received: January 20, 2020; Accepted: February 8, 2021

Abstract

In this paper, we define the G-Brauer algebras DfGx, where G is a cyclic group, called cyclic G-Brauer algebras, as the linear span of r-signed 1-factors and the generalized m,k signed partial 1-factors is to analyse the multiplication of basis elements in the quotient IfG(x,2k). Also, we define certain symmetric matrices Tm,k[λ](x) whose entries are indexed by generalized m,k signed partial 1-factor. We analyse the irreducible representations of DfGx by determining the quotient IfG(x,2k) of DfGx by its radical. We also find the eigenvalues and eigenspaces of Tm,k[λ](x) for some values of m and k using the representation theory of the generalised symmetric group. The matrices Tm,k[λ](x) whose entries are indexed by generalised m,k signed partial 1-factors, which helps in determining the non semisimplicity of these cyclic G-Brauer algebras DfGx, where G=r.

Keywords: G-Brauer algebras, centraliser algebras, eigenvalues