Semisimple symplectic characters of finite unitary groups

TL;DR

This paper characterizes certain irreducible complex characters of finite unitary groups, providing a count and a combinatorial formula for character values, advancing understanding of their representation theory.

Contribution

It establishes the exact number of irreducible characters with specific properties and derives a new combinatorial formula for character values at central elements.

Findings

Number of characters with degree not divisible by p and Frobenius-Schur indicator -1 is q^{m-1}.

Derived a combinatorial formula for character values at central elements.

Enhanced understanding of the representation theory of finite unitary groups.

Abstract

Let $G = {\rm U}(2m, {\mathbb F}_{q^2})$ be the finite unitary group, with $q$ the power of an odd prime $p$. We prove that the number of irreducible complex characters of $G$ with degree not divisible by $p$ and with Frobenius-Schur indicator -1 is $q^{m-1}$. We also obtain a combinatorial formula for the value of any character of ${\rm U}(n, {\mathbb F}_{q^2})$ at any central element, using the characteristic map of the finite unitary group.