The homology of the Brauer algebras

Abstract

This paper investigates the homology of the Brauer algebras, interpreted as appropriate $${{\,\mathrm{Tor}\,}}$$ 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 $$\delta $$ δ 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 $$\delta $$ δ is not invertible, this isomorphism still holds in a range of degrees that increases with n.

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“…For the stable homology, the left-hand side of this isomorphism is well known by the Barratt-Priddy-Quillen theorem [Barratt and Priddy 1972;Friedlander and Mazur 1994]. The above results exactly parallel the situation for the Brauer algebras, and as discussed in [Boyd et al 2021] are reminiscent of the relationship between S n and the automorphism groups of free groups Aut(F n ) (see Galatius [2011]).…”

“…In the last few years it has become increasingly apparent that the techniques of homological stability, which are most commonly applied to families of groups, can be successfully applied to families of algebras, where homology is interpreted as an appropriate Tor group. Indeed, Boyd and Hepworth [2020], Boyd, Hepworth and Patzt [2021], Hepworth [2022] and Moselle [2022] proved homological stability for Temperley-Lieb algebras, Brauer algebras, and Iwahori-Hecke algebras of types A and B respectively, and identified the stable homology in the first two cases. The Temperley-Lieb and Brauer algebras failed to satisfy a certain flatness condition that holds automatically for families of groups, necessitating the introduction of the new technique of inductive resolutions.…”

“…Using related techniques, Sroka [2023] showed that the homology of the Temperley-Lieb algebra on an odd number of strands vanishes in positive degrees, in contrast to the known nonvanishing for an even number of strands. More recently, Boyde [2022] used a careful study of idempotents to unify and generalise the "invertible parameter" results from [Boyd and Hepworth 2020;Boyd et al 2021], together with Sroka's vanishing result. In this paper, we prove homological stability for the partition algebras, and we identify their stable homology.…”

“…In Section 2 we introduce partition algebras and provide the necessary background needed for the rest of the paper. In Section 3 we restate an abstract form of the principle that lies behind the technique of inductive resolutions that was introduced in [Boyd and Hepworth 2020], and was a crucial ingredient in [Boyd and Hepworth 2020] and [Boyd et al 2021]. In Section 4 we establish the existence of inductive resolutions for the partition algebras.…”

Homological stability for Iwahori–Hecke algebras

“…To the best of our knowledge, the present paper and [2] are the first homological stability results of their kind for algebras that are not group algebras, and our hope is that they will serve as a proof of concept for the export of homological stability techniques into new algebraic contexts. Indeed, since the appearance of the present paper and [2], the author together with Boyd and Patzt have used the same set of techniques to study the homology of Brauer algebras [3]. On a related note, Sroka [38] has adapted techniques from the geometry and topology of Coxeter groups (specifically the Davis complex) to study the homology of odd Temperley–Lieb algebras.…”