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Multiplicity-free representations of the principal A1-subgroup in a simple algebraic group

Aluna Rizzoli, Donna Testerman

TL;DR

The paper classifies irreducible $kG$-modules whose restriction to a principal $A_1$-subgroup is multiplicity-free in positive characteristic, extending characteristic-zero results. It first handles rank-2 cases, proving MF occurs only for type $A_2$ with $p=3$ and $oldsymbol{ ho}= ext{omega}_1+ ext{omega}_2$, or type $B_2$ with $p=5$ and $oldsymbol{ ho}=2 ext{omega}_1$ (plus the $p$-adic lift structure in the general $p$-restricted setting). For higher rank ($ ext{rank}(G) ge 3$), the authors show that MF cannot occur when $pigle r$, by a detailed weight-space analysis and reduction across root systems, leaving only the rank-2 special cases as MF possibilities. They further extend the results to non-$p$-restricted highest weights via a $p$-adic decomposition and tensor-product argument, yielding a corollary that mirrors the rank-2 MF criteria in the unrestricted setting. The work provides a foundational step toward a full multiplicity-free subgroups program in positive characteristic and connects to existing results in the zero-characteristic and tilting-module literature. Key techniques include weight-space recurrences, dimension bounds $B(r)$, Premet-type weight comparisons, and careful case analysis across Dynkin types with $pigge h(G)$.

Abstract

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p>0. Here we classify all irreducible kG-modules for which the principal A1 has no repeated composition factors, extending the work of Liebeck-Seitz-Testerman which treated the same question when k is replaced by an algebraically closed field of characteristic zero.

Multiplicity-free representations of the principal A1-subgroup in a simple algebraic group

TL;DR

The paper classifies irreducible -modules whose restriction to a principal -subgroup is multiplicity-free in positive characteristic, extending characteristic-zero results. It first handles rank-2 cases, proving MF occurs only for type with and , or type with and (plus the -adic lift structure in the general -restricted setting). For higher rank (), the authors show that MF cannot occur when , by a detailed weight-space analysis and reduction across root systems, leaving only the rank-2 special cases as MF possibilities. They further extend the results to non--restricted highest weights via a -adic decomposition and tensor-product argument, yielding a corollary that mirrors the rank-2 MF criteria in the unrestricted setting. The work provides a foundational step toward a full multiplicity-free subgroups program in positive characteristic and connects to existing results in the zero-characteristic and tilting-module literature. Key techniques include weight-space recurrences, dimension bounds , Premet-type weight comparisons, and careful case analysis across Dynkin types with .

Abstract

Let G be a simple algebraic group defined over an algebraically closed field k of characteristic p>0. Here we classify all irreducible kG-modules for which the principal A1 has no repeated composition factors, extending the work of Liebeck-Seitz-Testerman which treated the same question when k is replaced by an algebraically closed field of characteristic zero.
Paper Structure (10 sections, 57 theorems, 28 equations)

This paper contains 10 sections, 57 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. The case where $G$ has rank $2$
  4. The case where $G$ is $A_2$
  5. The case where $G$ is $B_2$
  6. The case where $G$ is $G_2$
  7. The case where $G$ has rank at least $3$
  8. The case $c_ic_j$ not $0$
  9. The case where $\lambda = b\omega_i$
  10. Proof of Corollary \ref{['main corollary']}

Key Result

Theorem 1

Suppose that $\lambda$ is $p$-restricted. Then $L(\lambda)\downarrow A$ is $MF$ if and only if one of the following holds.

Theorems & Definitions (112)

  • Theorem 1
  • Corollary 2
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 102 more