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Classifying representations of finite classical groups of Lie type of dimension up to $\ell^4$

Luis Gutiérrez Frez, Adrian Zenteno

TL;DR

The article classifies irreducible representations $L(\lambda)$ of finite classical groups of Lie type with dimension bounded by $\ell^4/16$, splitting the analysis into type $A_{\ell}$ and the other types. It reduces to finitely many $p$-restricted highest weights via the Steinberg tensor product theorem and employs root-system/combinatorial methods alongside the Freudenthal framework to bound dimensions and multiplicities; for type $A_{\ell}$ with $\ell \ge 15$ it gives a complete admissible-weight list and explicit dimensions, while for types $B_\ell$, $C_\ell$, and $D_\ell$ it provides complete admissible-weight classifications (with detailed multiplicity data in the Appendix) and explicit dimension formulas under suitable prime conditions. The results yield concrete dimensions for several weights in large characteristic and enable conditional applications to automorphic Galois representations and the inverse Galois problem, including statements about large residual images and potential Galois realizations of classical groups. Overall, the work bridges detailed representation-theoretic classifications with number-theoretic applications in Galois theory.

Abstract

Let $G$ be a finite classical group of Lie type of rank $\ell$, defined over a field of characteristic $p>2$. In this work, we classify the irreducible representations of $G$ whose dimensions are bounded by a constant proportional to $\ell$, and splits into two cases according to $G$ is of type $A_{\ell}$ or not. Furthermore, we discuss explicit formulas for computing the dimensions of such representations. The motivation for this work arises, in part, from a desire to obtain new results on two classical problems concerning Galois representations: the large image conjecture for automorphic Galois representations and the inverse Galois problem. We conclude the paper by giving some remarks on potential implications in these addresses.

Classifying representations of finite classical groups of Lie type of dimension up to $\ell^4$

TL;DR

The article classifies irreducible representations of finite classical groups of Lie type with dimension bounded by , splitting the analysis into type and the other types. It reduces to finitely many -restricted highest weights via the Steinberg tensor product theorem and employs root-system/combinatorial methods alongside the Freudenthal framework to bound dimensions and multiplicities; for type with it gives a complete admissible-weight list and explicit dimensions, while for types , , and it provides complete admissible-weight classifications (with detailed multiplicity data in the Appendix) and explicit dimension formulas under suitable prime conditions. The results yield concrete dimensions for several weights in large characteristic and enable conditional applications to automorphic Galois representations and the inverse Galois problem, including statements about large residual images and potential Galois realizations of classical groups. Overall, the work bridges detailed representation-theoretic classifications with number-theoretic applications in Galois theory.

Abstract

Let be a finite classical group of Lie type of rank , defined over a field of characteristic . In this work, we classify the irreducible representations of whose dimensions are bounded by a constant proportional to , and splits into two cases according to is of type or not. Furthermore, we discuss explicit formulas for computing the dimensions of such representations. The motivation for this work arises, in part, from a desire to obtain new results on two classical problems concerning Galois representations: the large image conjecture for automorphic Galois representations and the inverse Galois problem. We conclude the paper by giving some remarks on potential implications in these addresses.
Paper Structure (15 sections, 13 theorems, 128 equations, 2 tables)

This paper contains 15 sections, 13 theorems, 128 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on representations of finite simple groups of Lie type
  3. Root Systems
  4. Representations
  5. Dimension and Multiplicity
  6. Representations of finite classical groups of type $A_\ell$
  7. Multiplicities and dimension in finite classical groups
  8. Type $B_\ell$
  9. Type $C_\ell$ and $D_\ell$
  10. Some dimension calculations and a potential application
  11. Type $C_\ell$ and $D_\ell$
  12. Automorphic Galois representations
  13. Apendix
  14. Type $C_\ell$
  15. Type $D_\ell$

Key Result

Theorem 2.2

Let $\lambda \in X$ be a dominant weight. Then, there exist $r \in \mathbb{N}$ and $p$-restricted dominant weights $\lambda_0, \cdots, \lambda_r \in X$ such that as $G(K)$-modules,

Theorems & Definitions (31)

  • Remark 2.1
  • Theorem 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Example 3.3
  • Theorem 3.4
  • proof
  • Example 4.1
  • ...and 21 more