On the semisimplicity of the decorated partial Brauer algebras

Abstract

The decorated Partial Brauer algebras are finite dimensional diagram algebras contain Brauer algebras, Partial Brauer algebras and the group algebras \(R\widetilde{S_{n}}\), where \(\widetilde{S_{n}}\) is the wreath product group \(\mathbb {Z}_{2}\wr S_{n}\) of \(\mathbb {Z}_{2}\) with \(S_{n}\). In this paper, we study the semisimplicity criterion of the decorated partial Brauer algebras using two functors F and G. In particular, we determine for which value of the parameters this algebra is semisimple. This result can be considered as a generalization of Hanlon–Wales conjecture on Brauer algebra.