Stable homology isomorphisms for the partition and Jones annular algebras

TL;DR

This paper establishes homology isomorphisms between Jones annular algebras and cyclic groups, as well as partition algebras and symmetric groups, extending known stability ranges.

Contribution

It proves new homology isomorphisms for Jones annular and partition algebras beyond their usual stability ranges.

Findings

Homology of Jones annular algebras is isomorphic to cyclic groups below a certain line.

Homology of partition algebras is isomorphic to symmetric groups below a certain line.

Results extend the stability range of these algebraic isomorphisms.

Abstract

We show that the homology of the Jones annular algebras is isomorphic to that of the cyclic groups below a line of gradient $\frac{1}{2}$. We also show that the homology of the partition algebras is isomorphic to that of the symmetric groups below a line of gradient 1, strengthening a result of Boyd-Hepworth-Patzt. Both isomorphisms hold in a range exceeding the stability range of the algebras in question. Along the way, we prove the usual odd-strand and invertible parameter results for the Jones annular algebras.