Values of characters sums for finite unitary groups

TL;DR

This paper derives explicit formulas for character sums in finite unitary groups, extending known results and providing probabilistic interpretations, with applications to conjugacy class evaluations.

Contribution

It develops explicit formulas for permutation character values in finite unitary groups and extends character sum results to broader conjugacy classes, including probabilistic aspects.

Findings

Explicit formulas for permutation characters of U(2n, F_{q^2}) over Sp(2n, F_q)

Extended character sum results to arbitrary conjugacy classes in finite unitary groups

Probabilistic interpretations of character sum formulas

Abstract

A known result for the finite general linear group $\GL(n,\FF_q)$ and for the finite unitary group $\U(n,\FF_{q^2})$ posits that the sum of the irreducible character degrees is equal to the number of symmetric matrices in the group. Fulman and Guralnick extended this result by considering sums of irreducible characters evaluated at an arbitrary conjugacy class of $\GL(n,\FF_q)$. We develop an explicit formula for the value of the permutation character of $\U(2n,\FF_{q^2})$ over $\Sp(2n,\FF_q)$ evaluated an an arbitrary conjugacy class and use results concerning Gelfand-Graev characters to obtain an analogous formula for $\U(n,\FF_{q^2})$ in the case where $q$ is an odd prime. These results are also given as probabilistic statements.