On Foulkes characters

TL;DR

This paper explores Foulkes characters of full monomial groups, providing orthogonality relations, solutions for their product decompositions, and new applications in Markov chains and enumeration of Riemann surfaces.

Contribution

It introduces new orthogonality relations, decompositions, and applications of Foulkes characters, including a novel proof of a generalized theorem by Zagier.

Findings

Orthogonality relations for Foulkes characters established

Three solutions to product decomposition problems provided

New applications in Markov chain analysis and Riemann surface enumeration

Abstract

Orthogonality relations for Foulkes characters of full monomial groups are presented, along with three solutions to the problem of decomposing products of these characters, and new applications, including a product reformulation of a Markov chain for adding random numbers studied by Diaconis and Fulman, and a new proof of a theorem of Zagier which generalizes one of Harer and Zagier on the enumeration of Riemann surfaces of a given genus.