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

Abstract

For a finite group $G$, let $\mathrm{AD}(G)$ denote the Fourier norm of the antidiagonal in $G\times G$. It was shown recently by the author (IMRN, 2023) that $\mathrm{AD}(G)$ 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 $\mathrm{AD}(G)$. As a by-product, we show that the set of values of $\mathrm{AD}(G)$ does not contain all its limit points. Some other calculations or bounds for $\mathrm{AD}(G)$ are given for familiar classes of finite groups. We also indicate a connection between $\mathrm{AD}(G)$ and the commuting probability of $G$, and use this to show that every finite group $G$ satisfying $\mathrm{AD}(G)<\frac{61}{15}$ must be solvable; here the value $\frac{61}{15}$ is best possible.

v4: AMS-LaTeX, 10pt mathpazo fonts; 18 pages. Various corrections and improvements, incorporating feedback from a referee; the main results are unchanged. Accepted for publication in J. Austral. Math. Soc

References in corpus (1)

• An explicit minorant for the amenability constant of the Fourier algebra