Cycle indices for finite orthogonal groups of even characteristic

TL;DR

This paper develops cycle index generating functions for finite orthogonal groups in even characteristic and explores their enumerative applications, involving character theory and probability measures on partitions.

Contribution

It introduces new cycle index formulas for orthogonal groups in even characteristic and analyzes character values at unipotent elements.

Findings

Derived explicit cycle index generating functions.

Determined complex character values at unipotent elements.

Explored probability measures on integer partitions.

Abstract

We develop cycle index generating functions for orthogonal groups in even characteristic, and give some enumerative applications. A key step is the determination of the values of the complex linear-Weil characters of the finite symplectic group, and their inductions to the general linear group, at unipotent elements. We also define and study several natural probability measures on integer partitions.