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

**Published online** March 31, 2022

Copyright © Kyungpook Mathematical Journal.

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

In this paper, we define the G-Brauer algebras ${D}_{f}^{G}\left(x\right)$, 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 ${\overrightarrow{{I}_{f}}}^{G}(x,2k)$. Also, we define certain symmetric matrices ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$ whose entries are indexed by generalized m,k signed partial 1-factor. We analyse the irreducible representations of ${D}_{f}^{G}\left(x\right)$ by determining the quotient ${\overrightarrow{{I}_{f}}}^{G}(x,2k)$ of ${D}_{f}^{G}\left(x\right)$ by its radical. We also find the eigenvalues and eigenspaces of ${\overrightarrow{T}}_{m,k}^{[\lambda ]}(x)$ for some values of m and k using the representation theory of the generalised symmetric group. The matrices ${T}_{m,k}^{[\lambda ]}(x)$ 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 ${D}_{f}^{G}\left(x\right)$, where $G={\mathbb{Z}}_{r}$.

**Keywords**: *G*-Brauer algebras, centraliser algebras, eigenvalues