Cohomology of Tanabe algebras

TL;DR

This paper demonstrates that the (co)homology of Tanabe algebras is isomorphic to that of symmetric groups, independent of parameters or parity, revealing a novel invariant property in algebraic cohomology.

Contribution

It establishes the first known case where the (co)homology of a diagram algebra is independent of parameters and parity, linking Tanabe algebra cohomology to symmetric groups.

Findings

Tanabe algebra (co)homology is isomorphic to symmetric group (co)homology.

(Co)homology independence from parameters and parity.

Includes study of uniform block permutation and totally propagating partition 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.