Generalized rook-Brauer algebras and their homology

TL;DR

This paper generalizes rook-Brauer algebras by incorporating structured, labeled, and braided diagrams, then studies their homology, revealing connections to group homology and establishing stability results.

Contribution

It introduces a new class of generalized diagram algebras with equivariance and braiding, and analyzes their homology relating to braid and semi-direct product groups.

Findings

Homology of generalized algebras identified with group homology of braid groups and semi-direct products.

Homological stability results established for these generalized diagram algebras.

Homology of certain diagrams with odd edges matches group homology without parameter restrictions.

Abstract

Rook-Brauer algebras are a family of diagram algebras. They contain many interesting subalgebras: rook algebras, Brauer algebras, Motzkin algebras, Temperley-Lieb algebras and symmetric group algebras. In this paper, we generalize the rook-Brauer algebras and their subalgebras by allowing more structured diagrams. We introduce equivariance by labelling edges of a diagram with elements of a group $G$. We introduce braiding by insisting that when two strands cross, they do so as either an under-crossing or an over-crossing. We also introduce equivariant, braided diagrams by combining these structures. We then study the homology of our diagram algebras, as pioneered by Boyd and Hepworth, using methods introduced by Boyde. We show that, given certain invertible parameters, we can identify the homology of our generalized diagram algebras with the group homology of the braid groups $B_n$ and the semi-direct products $G^n\rtimes \Sigma_n$ and $G^n\rtimes B_n$. This allows us to deduce homological stability results for our generalized diagram algebras. We also prove that for diagrams with an odd number of edges, the homology of equivariant Brauer algebras and equivariant Temperley-Lieb algebras can be identified with the group homology of $G^n\rtimes \Sigma_n$ and $G^n$ respectively, without any conditions on parameters.