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Permutation modules of the walled Brauer algebras

Sulakhana Chowdhury, Geetha Thangavelu

TL;DR

The paper addresses the decomposition of permutation modules for the direct product of symmetric groups $K\\mathfrak{S}_{a,b}$ and the walled Brauer algebras $\\mathcal{B}_{r,t}(\\delta)$ in characteristic not equal to $2$ or $3$. It develops a dual Specht filtration framework and shows that permutation modules admit a direct sum decomposition into indecomposable Young modules, with a uniquely determined top summand, using cellular stratification and induction techniques. The main result provides an explicit decomposition formula $M(l,(\\lambda,\\mu))\\cong Y(l,(\\lambda,\\mu))\\oplus \bigoplus a_{(m,(\\lambda',\\mu'))} Y(m,(\\lambda',\\mu'))$ ordered by a dominance relation, and proves that the Young modules are relative projective covers of cell modules inside a standardly stratified setting. The work also furnishes a Schur-algebraic perspective for the walled Brauer context, establishing a foundation for further structural and homological analysis of these algebras and their module categories.

Abstract

In this article, we study the permutation modules and Young modules of the group algebras of the direct product of symmetric groups $K\mathfrak{S}_{a,b}$, and the walled Brauer algebras $\B_{r,t}(δ)$. In the category of dual Specht-filtered modules, if the characteristic of the field is neither $2$ nor $3$, then the permutation modules are dual Specht filtered, and the Young modules are relative projective cover of the dual Specht modules. We prove that the restriction of the cell modules of $\B_{r,t}(δ)$ to the group algebras of the direct product of the symmetric groups is dual Specht filtered, and the Young modules act as the relative projective cover of the cell modules of $\B_{r,t}(δ)$. Finally, we prove that if $\mathrm{char}~K \neq 2,3$, then the permutation module of $\B_{r,t}(δ)$ can be written as a direct sum of indecomposable Young modules.

Permutation modules of the walled Brauer algebras

TL;DR

The paper addresses the decomposition of permutation modules for the direct product of symmetric groups and the walled Brauer algebras in characteristic not equal to or . It develops a dual Specht filtration framework and shows that permutation modules admit a direct sum decomposition into indecomposable Young modules, with a uniquely determined top summand, using cellular stratification and induction techniques. The main result provides an explicit decomposition formula ordered by a dominance relation, and proves that the Young modules are relative projective covers of cell modules inside a standardly stratified setting. The work also furnishes a Schur-algebraic perspective for the walled Brauer context, establishing a foundation for further structural and homological analysis of these algebras and their module categories.

Abstract

In this article, we study the permutation modules and Young modules of the group algebras of the direct product of symmetric groups , and the walled Brauer algebras . In the category of dual Specht-filtered modules, if the characteristic of the field is neither nor , then the permutation modules are dual Specht filtered, and the Young modules are relative projective cover of the dual Specht modules. We prove that the restriction of the cell modules of to the group algebras of the direct product of the symmetric groups is dual Specht filtered, and the Young modules act as the relative projective cover of the cell modules of . Finally, we prove that if , then the permutation module of can be written as a direct sum of indecomposable Young modules.

Paper Structure

This paper contains 16 sections, 29 theorems, 44 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Walled Brauer algebras
  3. Cellularly stratification of $\mathcal{B}_{r,t}(\delta)$
  4. Permutation modules of $K\mathfrak{S}_{a,b}$
  5. Representation of symmetric group $\mathfrak{S}_a$
  6. Representation theory of $K\mathfrak{S}_{a,b}$
  7. Permutation module of $\mathcal{B}_{r,t}(\delta)$
  8. Induction and Restriction functor
  9. Young modules
  10. Main Theorem
  11. Cell filtration
  12. Stabilizer of $v$ in $V_l$
  13. The structure of $J_me_l$
  14. Cell filtration of $M(l,(\lambda,\mu))$
  15. Relative Projectivity
  16. ...and 1 more sections

Key Result

Theorem 1.1

If $(l,(\lambda,\mu)) \in \Lambda$ and $\mathrm{char}~K \neq 2,3$, then the decomposition of the permutation module $M(l,(\lambda,\mu))$ is given by where $(m,(\lambda',\mu'))$ runs over all $(m,(\lambda',\mu')) \leq (l,(\lambda,\mu))$, $a_{(m,(\lambda',\mu'))}:=[M(l,(\lambda,\mu)):Y(m,(\lambda',\mu'))]$ is the multiplicity of $Y(m,(\lambda',\mu'))$ in $M(l,(\lambda,\mu))$, and $Y(l,(\lambda,\mu

Figures (5)

  • Figure 1: Generators of the walled Brauer algebra.
  • Figure 2: Idempotents of $\mathcal{B}_{r,t}(\delta)$ for $\delta \neq 0$.
  • Figure 3: Idempotents of $\mathcal{B}_{r,t}(\delta)$ for $\delta= 0$ and when one of $r$ or $t$ is at least $2$.
  • Figure 4: Identify a partial diagram $v$ as an element of $\mathfrak{S}_l$.
  • Figure 5: An element of $\mathop{\mathrm{Stab}}\nolimits_{K\mathfrak{S}_{l,l}}(v).$

Theorems & Definitions (58)

  • Theorem 1.1
  • Definition 2.1: HHKP, Definition 2.1
  • Proposition 2.2: CVDM, Proposition 2.6
  • Theorem 2.3
  • Theorem 3.1: Ja, Theorem 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 48 more