Homology of Rook-Brauer Algebras and Motzkin Algebras

TL;DR

This paper proves that the homology of Rook-Brauer and Motzkin algebras exhibits stability properties, with Rook-Brauer's homology matching that of symmetric groups and Motzkin's homology vanishing in positive degrees, under certain conditions.

Contribution

It introduces an inductive resolution technique to establish homological stability for Rook-Brauer and Motzkin algebras, extending understanding of their algebraic properties.

Findings

Homology of Rook-Brauer algebra is isomorphic to that of symmetric groups.

Homology of Motzkin algebra vanishes in positive degrees.

Homological stability is established for both algebras.

Abstract

Using the technique of inductive resolution introduced in arXiv:2303.07979, we prove that the homology of Rook-Brauer Algebra, interpreted as appropriate Tor-group, is isomorphic to that of symmetric group for all degrees under the assumption that $\epsilon$ in $R$ is invertible; furthermore, we also prove the homology of the Motzkin algebras vanishes in positive degrees under the same assumption. These results thereby establish homological stability of both algebras.