Revisiting Foulkes characters of wreath products

TL;DR

This paper explores the structure of Foulkes characters in wreath products, introducing new dualities and constructions that deepen understanding of their properties and representations.

Contribution

It introduces two specialized dualities for wreath products, providing new decompositions and constructions of Foulkes characters, and offers alternative proofs of their properties.

Findings

Foulkes characters are the extreme rays of the cone of block characters.

Constructs representations with traces equal to Foulkes characters.

Provides new dualities and decompositions for wreath product characters.

Abstract

The article is concerned with the Foulkes characters of wreath products, which are block characters of wreath products, i.e., the positive-definite class functions depending only on the length of its elements. Inspired by the works of Gnedin--Gorin--Kerov and Miller, we introduce two specializations of the Schur--Weyl--Sergeev duality for wreath products and obtain two families of block characters, which provide a decomposition and an alternative construction of the Foulkes characters of wreath products. In particular, we give alternative proofs on some remarkable properties of the Foulkes characters. Along the way, we show that the Foulkes characters are the extreme rays of the cone of the block characters of wreath products and construct the representations with traces being the Foulkes characters via the coinvariant algebra of wreath products.