Cohomology of Tanabe algebras
Andrew Fisher, Daniel Graves
TL;DR
The paper addresses the (co)homology of Tanabe algebras, a family of partition-algebra subalgebras tied to Schur–Weyl dualities with complex reflection groups, and proves their (co)homology is isomorphic to that of the symmetric groups, independent of the parameter $\delta$ and the parity of $n$. It develops a framework using $k$-free idempotent left covers and Mayer–Vietoris complexes to transfer (co)homology from a quotient $A/I \cong k[\Sigma_n]$ to the ambient algebras, enabling stable identifications for Tanabe algebras, uniform block algebras, totally propagating partition algebras, and, via known connections, partition/Jones annular algebras. The main results show that $\mathrm{Tor}$ and $\mathrm{Ext}$ with the trivial module coincide with the homology and cohomology of $\Sigma_n$, respectively, in a manner that is independent of $\delta$ and $n$, and they establish cohomological stability for partition algebras. These findings provide the first parameter- and parity-independent cohomology identifications in this diagram-algebra setting and reveal deep links to symmetric-group (co)homology with broad applicability to related algebras.
Abstract
In this paper we study the (co)homology of Tanabe algebras, which are a family of subalgebras of the partition algebras exhibiting a Schur-Weyl duality with certain complex reflection groups. The homology of the partition algebras has been shown to be related to the homology of the symmetric groups by Boyd-Hepworth-Patzt and the results they obtain depend on a parameter. In all known results, the homology of a diagram algebra is dependent on one of two things: the invertibility of a parameter in the ground ring or the parity of the positive integer indexing the number of pairs of vertices. We show that the (co)homology of Tanabe algebras is isomorphic to the (co)homology of the symmetric groups and that this is independent of both the parameter and the parity of the index. To the best of our knowledge, this is the first example of a result of this sort. Along the way we will also study the (co)homology of the uniform block permutation algebra and the totally propagating partition algebra as well collecting cohomological analogues of known results for the homology of partition algebras and Jones annular algebras.