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Small values and forbidden values for the Fourier antidiagonal constant of a finite group

Yemon Choi

TL;DR

This paper determines the small-value behavior of the Fourier antidiagonal constant ${\rm AD}(G)$ for finite groups, tying it to irreducible character degrees via ${\rm AD}(G)=\frac{1}{|G|}\sum_{\varphi\in{\rm Irr}(G)} d(\varphi)^3$ and establishing a precise range in $[1,2]$: ${\rm AD}(G)\in\{2-1/n: n\in\mathbb{N}\}$ when ${\rm AD}(G)\le 2$. It proves a new universal lower bound ${\rm AD}(G)\ge 2+|G'|^{-1}$ under a degree-3 condition, and a sharp solvability threshold ${\rm AD}(G)<\frac{61}{15}$ implying solvability, with ${\rm AD}(A_5)=\frac{61}{15}$ demonstrating sharpness. The work combines an ${\mathcal L}$-orbit method with commuting-probability techniques, and, for the non-solvable case, a refined Tong-Viet approach aided by classifications of simple groups with small degree representations. It provides extensive exact computations of ${\rm AD}(G)$ for families such as extraspecial $p$-groups, affine groups ${\mathbb F}_q\rtimes {\mathbb F}_q^{\times}$, ${\rm SL}(2,q)$, and several quasi-simple and perfect groups, yielding a spectrum of values and highlighting the links between representation-theoretic data and amenability-type constants. The results reveal a meaningful structure: small values force $G$ toward abelian- or solvable-like behavior, while the non-solvable case is tightly constrained by the commuting probability and low-degree representations.

Abstract

For a finite group $G$, let ${\rm AD}(G)$ denote the Fourier norm of the antidiagonal in $G\times G$. It was shown recently by the author (IMRN, 2023) that ${\rm AD}(G)$ coincides with the amenability constant of the Fourier algebra of $G$, and is equal to the normalized sum of the cubes of character degrees of $G$. Motivated by a gap result for amenability constants due to Johnson (JLMS, 1994), we determine exactly which numbers in the interval $[1,2]$ arise as values of ${\rm AD}(G)$. As a by-product, we show that the set of values of ${\rm AD}(G)$ does not contain all its limit points. Some other calculations or bounds for ${\rm AD}(G)$ are given for familiar classes of finite groups. We also indicate a connection between ${\rm AD}(G)$ and the commuting probability of $G$, and use this to show that every finite group $G$ satisfying ${\rm AD}(G)< \frac{61}{15}$ must be solvable; here the value $\frac{61}{15}$ is best possible.

Small values and forbidden values for the Fourier antidiagonal constant of a finite group

TL;DR

This paper determines the small-value behavior of the Fourier antidiagonal constant for finite groups, tying it to irreducible character degrees via and establishing a precise range in : when . It proves a new universal lower bound under a degree-3 condition, and a sharp solvability threshold implying solvability, with demonstrating sharpness. The work combines an -orbit method with commuting-probability techniques, and, for the non-solvable case, a refined Tong-Viet approach aided by classifications of simple groups with small degree representations. It provides extensive exact computations of for families such as extraspecial -groups, affine groups , , and several quasi-simple and perfect groups, yielding a spectrum of values and highlighting the links between representation-theoretic data and amenability-type constants. The results reveal a meaningful structure: small values force toward abelian- or solvable-like behavior, while the non-solvable case is tightly constrained by the commuting probability and low-degree representations.

Abstract

For a finite group , let denote the Fourier norm of the antidiagonal in . It was shown recently by the author (IMRN, 2023) that coincides with the amenability constant of the Fourier algebra of , and is equal to the normalized sum of the cubes of character degrees of . Motivated by a gap result for amenability constants due to Johnson (JLMS, 1994), we determine exactly which numbers in the interval arise as values of . As a by-product, we show that the set of values of does not contain all its limit points. Some other calculations or bounds for are given for familiar classes of finite groups. We also indicate a connection between and the commuting probability of , and use this to show that every finite group satisfying must be solvable; here the value is best possible.
Paper Structure (16 sections, 31 theorems, 41 equations)

This paper contains 16 sections, 31 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Background context
  3. Our main new results
  4. Outline of this paper
  5. Some easy lower bounds on $\mathop{\mathrm{AD}}\nolimits$
  6. The proof of Proposition \ref{['p:new lower bound']}
  7. Further examples and implications of small values
  8. Values of $\mathop{\mathrm{AD}}\nolimits$ for particular groups
  9. Structural consequences for $G$ of upper bounds on $\mathop{\mathrm{AD}}\nolimits$
  10. A sharp lower bound on $\mathop{\mathrm{AD}}\nolimits$ for non-solvable groups
  11. Easier proofs of some known results
  12. Finite groups with two character degrees and derived subgroup of order $2$
  13. A self-contained proof of Lemma \ref{["l:|G'|=2"]}
  14. A proof of Lemma \ref{['l:annoying']} with $\delta_0= 1/20$
  15. The case $d=3$.
  16. ...and 1 more sections

Key Result

Proposition 1.1

Let $G$ be a finite group, and suppose that $d({\varphi})\leq 2$ for all $\varphi\in\mathop{\mathrm{Irr}}\nolimits(G)$. Then $\mathop{\mathrm{AD}}\nolimits(G) \in \{ 2- n^{-1} \colon n\in\mathbb N\}$.

Theorems & Definitions (69)

  • Proposition 1.1: implicitly folklore
  • Theorem 1.2: The possible values of $\mathop{\mathrm{AD}}\nolimits(G)$ in $[1,2]$
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5: A threshold ensuring solvability
  • Proposition 1.6
  • proof : Proof of Proposition \ref{['p:if maxdeg leq 2']}
  • Example 2.1: Dihedral groups
  • Lemma 2.2: An all-purpose lower bound
  • proof
  • ...and 59 more