Character Theory of Monoids over an Arbitrary Field

TL;DR

This paper extends the character theory of finite monoids from complex numbers to arbitrary fields, providing new proofs and applications in symbolic dynamics and language theory.

Contribution

It generalizes the character theory of finite monoids to any field, enabling broader applications and a new proof of a key theorem in symbolic dynamics.

Findings

Character theory over arbitrary fields is analogous to the complex case.

The characteristic function of a regular cyclic language is a virtual character.

New proof of the rationality of the zeta function of a sofic shift.

Abstract

The basic character theory of finite monoids over the complex numbers was developed in the sixties and seventies based on work of Munn, Ponizovsky, McAlister, Rhodes and Zalcstein. In particular, McAlister determined the space of functions spanned by the irreducible characters of a finite monoid over $\mathbb C$ and the ring of virtual characters. In this paper, we present the corresponding results over an arbitrary field. As a consequence, we obtain a quick proof of the theorem of Berstel and Reutenauer that the characteristic function of a regular cyclic language is a virtual character of the free monoid. This is a crucial ingredient in their proof of the rationality of the zeta function of a sofic shift in symbolic dynamics.