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Comparing cohomology via exact split pairs in diagram algebras

Sulakhana Chowdhury, Geetha Thangavelu

TL;DR

The paper develops a unified method to compare cohomology across module categories of diagram algebras by establishing exact split pairs between diagram algebras and their input (corner) algebras. Using the framework of cellular algebras and iterated inflations, it proves that for $A$-Brauer, cyclotomic Brauer, and walled Brauer algebras one can realize $D_n(A)$ or $\mathcal{B}_n^r(\pmb{\delta})$ as iterated inflations with corner split quotients $A\wr \mathfrak{S}_{n-2l}$ or $\mathcal{H}_{n-2l}^r$, yielding explicit exact split pairs via induction/restriction functors $\text{ind}_l, \text{Res}_l$. Consequently, Hom and Ext between cell modules reduce to those for the input algebras, enabling precise non-vanishing criteria and global-dimension analyses; finite global dimension results follow under standard modular conditions (e.g., $p>n$ or $p=0$) and invertible delta parameters. The work extends cohomology comparison to several families of diagram algebras, with concrete corollaries for cyclotomic and walled Brauer contexts and implications for quasi-hereditary structure.

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.

Comparing cohomology via exact split pairs in diagram algebras

TL;DR

The paper develops a unified method to compare cohomology across module categories of diagram algebras by establishing exact split pairs between diagram algebras and their input (corner) algebras. Using the framework of cellular algebras and iterated inflations, it proves that for -Brauer, cyclotomic Brauer, and walled Brauer algebras one can realize or as iterated inflations with corner split quotients or , yielding explicit exact split pairs via induction/restriction functors . Consequently, Hom and Ext between cell modules reduce to those for the input algebras, enabling precise non-vanishing criteria and global-dimension analyses; finite global dimension results follow under standard modular conditions (e.g., or ) and invertible delta parameters. The work extends cohomology comparison to several families of diagram algebras, with concrete corollaries for cyclotomic and walled Brauer contexts and implications for quasi-hereditary structure.

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 -Brauer algebras, cyclotomic Brauer algebras, and walled Brauer algebras with their respective input algebras.

Paper Structure

This paper contains 10 sections, 17 theorems, 17 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The $A$-Brauer algebra
  4. Exact split pair ($D_n(A),A\wr \mathfrak{S}_n$)
  5. The Cyclotomic Brauer algebra
  6. The Generalized symmetric group
  7. Exact split pair ($\mathcal{B}_n^r(\pmb{\delta}), \mathcal{H}_n^r$)
  8. The Walled Brauer algebra
  9. Cellular structure
  10. Exact split pair ($\mathcal{B}_{r,t}(\delta), R\mathfrak{S}_{r,t}$)

Key Result

Theorem 2.1

Let $A$ and $B$ be any arbitrary rings and $S$ a left $B$-module. If $B$ is a corner split quotient of $A$. Then the functors $F = - \otimes_A Ae$ and $G =- \otimes_B B \otimes_{eAe} S$ form an exact split pair.

Figures (6)

  • Figure 1: Generators of the $A$-Brauer algebra.
  • Figure 2: Idempotent of the $A$-Brauer algebra for an invertible $\delta$.
  • Figure 3: Idempotent of the $A$-Brauer algebra when $\delta=0$, and $n$ is odd.
  • Figure 4: Generators of the walled Brauer algebra.
  • Figure 5: Idempotent of $\mathcal{B}_{r,t}(\delta)$ for invertible $\delta$.
  • ...and 1 more figures

Theorems & Definitions (36)

  • Theorem 2.1: DK, Lemma 3.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 26 more