Comparing cohomology via exact split pairs in diagram algebras

TL;DR

This paper investigates the relationship between cohomology in module categories of diagram algebras and their input algebras, establishing conditions for exact split pairs in specific Brauer algebra cases.

Contribution

It provides a new sufficient condition for the existence of exact split pairs between module categories of diagram algebras and input algebras, extending previous work.

Findings

Exact split pairs exist in A-Brauer, cyclotomic Brauer, and walled Brauer algebras.

The paper generalizes conditions for cohomology comparison in diagram algebras.

Results facilitate understanding of module category relationships in algebraic structures.

Abstract

In this article, we compare the cohomology between the categories of modules of the diagram algebras and the categories of modules of its input algebras. Our main result establishes a sufficient condition for exact split pairs between these two categories, analogous to a work by Diracca and Koenig in [7]. To be precise, we prove the existence of the exact split pairs in $A$-Brauer algebras, cyclotomic Brauer algebras, and walled Brauer algebras with their respective input algebras.