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Growth Problems for Representations of Finite Groups

David He

TL;DR

This work establishes a general framework for the growth of indecomposable summands in tensor powers of finite-group representations over arbitrary characteristic. It introduces a leading asymptotic growth rate a(n) expressed via Brauer data and a projective-cell fusion-graph approach, and proves an exact b(n) formula in characteristic zero. A key independence result shows that, for faithful V, the asymptotics depend only on center-related structure, enabling uniform descriptions across quotients and families with cyclic centers. The paper provides extensive, concrete computations for dihedral, symmetric, SL(2,q), cyclic, and Klein-four groups, plus nonsemisimple modular cases, illustrating convergence rates, variance bounds, and the role of second-largest eigenvalues; accompanying Magma code supports replication and exploration.

Abstract

We give a general asymptotic formula for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero we obtain additional results, including an exact formula for the growth rate. We compute various examples and also provide code used to compute our formulas.

Growth Problems for Representations of Finite Groups

TL;DR

This work establishes a general framework for the growth of indecomposable summands in tensor powers of finite-group representations over arbitrary characteristic. It introduces a leading asymptotic growth rate a(n) expressed via Brauer data and a projective-cell fusion-graph approach, and proves an exact b(n) formula in characteristic zero. A key independence result shows that, for faithful V, the asymptotics depend only on center-related structure, enabling uniform descriptions across quotients and families with cyclic centers. The paper provides extensive, concrete computations for dihedral, symmetric, SL(2,q), cyclic, and Klein-four groups, plus nonsemisimple modular cases, illustrating convergence rates, variance bounds, and the role of second-largest eigenvalues; accompanying Magma code supports replication and exploration.

Abstract

We give a general asymptotic formula for the growth rate of the number of indecomposable summands in the tensor powers of representations of finite groups, over a field of arbitrary characteristic. In characteristic zero we obtain additional results, including an exact formula for the growth rate. We compute various examples and also provide code used to compute our formulas.
Paper Structure (16 sections, 17 theorems, 31 equations, 13 figures, 1 table)

This paper contains 16 sections, 17 theorems, 31 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Growth problems
  3. Main results
  4. Examples
  5. Proof of main results
  6. Exact and asymptotic formulas
  7. Independence of asymptotic formula
  8. Examples in characteristic zero
  9. Dihedral groups
  10. Symmetric groups
  11. Special linear groups $\textnormal{SL}(2,q)$
  12. A family of semidirect products
  13. Nonsemisimple examples
  14. Special and general linear groups $\textnormal{SL}(2,q)$ and $\textnormal{GL}(2,q)$
  15. Cyclic groups
  16. ...and 1 more sections

Key Result

Theorem 1

Let $V$ be a faithful $kG$-module.

Figures (13)

  • Figure 1: Left: the ratio $b(n)/a(n)$ for faithful irreducible $\mathbb{C} D_{12}$-modules. Right: the variance $\lvert b(n)-a(n)\rvert$.
  • Figure 2: Left: the ratio $b(n)/a(n)$ for faithful irreducible $\mathbb{C} S_5$-modules, labelled with dimensions. Right: the character table for $S_5$ obtained from Magma.
  • Figure 3: Variance for the six-dimensional irreducible $\mathbb{C} S_5$-module. The $y$-axis uses a logarithmic scale.
  • Figure 4: Left: The ratio ${b(n)}/{a(n)}$ for all faithful irreducible $\mathbb{C} \text{SL}(2,5)$-modules, labelled with their dimensions. Right: the character table for $\text{SL}(2,5)$ from Magma, where if we let $\omega = \exp({2\pi i / 5})$ the symbols $Z1,Z1\#2$ correspond to $\omega+\omega^4$ and $\omega^2+\omega^3$ respectively.
  • Figure 5: $C_{32}\rtimes C_4$
  • ...and 8 more figures

Theorems & Definitions (41)

  • Theorem 1
  • Remark 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Remark 2
  • Theorem 5
  • Example 6
  • Lemma 7
  • proof
  • ...and 31 more