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On self-extensions of irreducible modules over symmetric groups, II

Lucia Morotti

TL;DR

The paper investigates self-extensions of irreducible representations $D^ u$ of symmetric groups over fields of odd characteristic $p\ge3$, aiming to prove $ ext{Ext}^1(D^ u,D^ u)=0$ in broader cases. It introduces $p$-separated partitions and uses abacus combinatorics and translation functors to extend vanishing results beyond previously known RoCK-block and low-weight cases. Key contributions include a RoCK-block–oriented reduction via abacus runner swaps (preserving $p$-separation) and new reduction theorems that bound Hom-spaces for translation functors and restrict potential counterexamples to a uniform residue pattern of normal nodes. Collectively, these results push toward the conjecture and supply new tools for analyzing modular representations of ${ m S}_n$ through core-quotient theory and abacus techniques.

Abstract

It is conjectured that irreducible representations of symmetric groups have no non-trivial self-extension over fields of odd characteristic. We improve on partial results showing evidence of this conjecture.

On self-extensions of irreducible modules over symmetric groups, II

TL;DR

The paper investigates self-extensions of irreducible representations of symmetric groups over fields of odd characteristic , aiming to prove in broader cases. It introduces -separated partitions and uses abacus combinatorics and translation functors to extend vanishing results beyond previously known RoCK-block and low-weight cases. Key contributions include a RoCK-block–oriented reduction via abacus runner swaps (preserving -separation) and new reduction theorems that bound Hom-spaces for translation functors and restrict potential counterexamples to a uniform residue pattern of normal nodes. Collectively, these results push toward the conjecture and supply new tools for analyzing modular representations of through core-quotient theory and abacus techniques.

Abstract

It is conjectured that irreducible representations of symmetric groups have no non-trivial self-extension over fields of odd characteristic. We improve on partial results showing evidence of this conjecture.

Paper Structure

This paper contains 8 sections, 13 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Submodule structure
  4. Partitions and abaci
  5. Representations of symmetric groups
  6. Translation functors
  7. $p$-separated partitions
  8. Reduction results

Key Result

Theorem A

Let $p\geq 3$, $\alpha\in{\mathscr {P}}_n$ and $\lambda\in{\mathscr P}^{\text{\rm $p$-reg}}_n$. If $\alpha$ is $p$-separated and $[S^\alpha:D^\lambda]>0$ then $\mathrm{Ext}^1(D^\lambda,D^\lambda)=0$.

Theorems & Definitions (22)

  • Conjecture
  • Theorem A
  • Theorem B
  • Theorem C
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • ...and 12 more