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eISSN 0454-8124
pISSN 1225-6951

### Article

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

Published online March 31, 2022

### 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 $If→G(x,2k)$. Also, we define certain symmetric matrices $T→m,k[λ](x)$ whose entries are indexed by generalized m,k signed partial 1-factor. We analyse the irreducible representations of $DfGx$ by determining the quotient $If→G(x,2k)$ of $DfGx$ by its radical. We also find the eigenvalues and eigenspaces of $T→m,k[λ](x)$ 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 $G=ℤr$.

Keywords: G-Brauer algebras, centraliser algebras, eigenvalues