Enumeration of Commuting Pairs in Lie Algebras over Finite Fields

TL;DR

This paper provides a new proof and generalizations of the generating function counting commuting pairs in Lie algebras over finite fields, extending previous results to unitary and symplectic cases and analyzing asymptotic behavior.

Contribution

It introduces a novel proof of the Feit-Fine generating function and extends it to Lie algebras of finite unitary and symplectic groups, also deriving new generating functions for nilpotent elements.

Findings

New proof of Feit-Fine generating function

Generalization to unitary and symplectic Lie algebras

Asymptotic analysis of commuting pairs

Abstract

Feit and Fine derived a generating function for the number of ordered pairs of commuting n by n matrices over the finite field F_q. This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas theory. In this note we give a new proof of the Feit-Fine result, and generalize it to the Lie algebra of finite unitary groups and to the Lie algebra of odd characteristic finite symplectic groups. We extract some asymptotic information from these generating functions. Finally, we derive generating functions for the number of commuting nilpotent elements for the Lie algebras of the finite general linear and unitary groups, and of odd characteristic symplectic groups.