Absolute values and tensor powers of irreducible characters

Abstract

Let $ χ$ be a character of a complex irreducible representation of a finite group $G$. We present a simple formula for the expectation of the random variable $(|χ|/χ(1))^{t} $ in terms of character ratios $ (|χ(g)|/χ(1))^{t}, \; g \in G, \; t \geq 0 $. As a follow up we briefly discuss asymptotic properties of the formula and its relation to the growth of dimensions of isotypic components in (virtual) tensor powers of irreducible representations

Figures (1)

• Figure 1: Graph of multiplicity of $\chi'_3$ in $(|\chi_3|/\chi_3(1))^t, \; t \geq 0$ (cf. Example 2.1). The graph is produced by Julia Plots package (cf. JuliaPlots). It seems that an interesting part of the graph is located within the interval $0 \leq t \leq 2$. The effect of $\chi'_3$ on $(|\chi_3|/\chi_3(1))^t$ is zero at both ends of the interval and is negative in its interior. The effect of $\chi'_3$ on the "tensor power" $(|\chi_3|/\chi_3(1))^t$ becomes positive when $t$ exceeds $2$

Theorems & Definitions (20)

• Remark 1.1
• Remark 1.2
• Lemma 1.1
• Remark 1.3
• Theorem 1.1
• Corollary 1.1
• Corollary 1.2
• Corollary 1.3
• Example 1.1
• Lemma 2.1