The homology of the partition algebras

Rachael Boyd, Richard Hepworth and Peter Patzt

Vol. 327 (2023), No. 1, 1–27

DOI: 10.2140/pjm.2023.327.1

Abstract

We show that the homology of the partition algebras, interpreted as appropriate $\mathrm{Tor}<!--FUNCTION\; APPLICATION-->$-groups, is isomorphic to that of the symmetric groups in a range of degrees that increases with the number of nodes. Further, we show that when the defining parameter $\delta$ of the partition algebra is invertible, the homology of the partition algebra is in fact isomorphic to the homology of the symmetric group in all degrees. These results parallel those obtained for the Brauer algebras in the authors’ earlier work, but with significant differences and difficulties in the inductive resolution and high acyclicity arguments required to prove them. Our results join the growing literature on homological stability for algebras, which now encompasses the Temperley–Lieb, Brauer and partition algebras, as well as the Iwahori–Hecke algebras of types $A$ and $B$.

Keywords

homology, homological stability, partition algebras

Mathematical Subject Classification

Primary: 16E40, 20J06

Secondary: 20B30

Milestones

Received: 20 June 2023

Revised: 19 October 2023

Accepted: 5 December 2023

Published: 18 February 2024