The homology of the Brauer algebras

TL;DR

This paper explores the homology of Brauer algebras, revealing its close relationship with symmetric group homology, especially when the defining parameter is invertible or in certain non-invertible cases.

Contribution

It establishes an isomorphism between the homology of Brauer algebras and symmetric groups under specific conditions, extending understanding of their algebraic structures.

Findings

Homology of Brauer algebra is isomorphic to symmetric group homology when the parameter is invertible.

The isomorphism extends to a range of degrees even when the parameter is not invertible.

Results connect algebraic properties of Brauer algebras with classical symmetric group homology.

Abstract

This paper investigates the homology of the Brauer algebras, interpreted as appropriate Tor-groups, and shows that it is closely related to the homology of the symmetric group. Our main results show that when the defining parameter of the Brauer algebra is invertible, then the homology of the Brauer algebra is isomorphic to the homology of the symmetric group, and that when the parameter is not invertible, this isomorphism still holds in a range of degrees that increases with n.