The Hopf structure of symmetric group characters as symmetric functions

Rosa Orellana,Mike Zabrocki

doi:10.5802/alco.170

Abstract

In [24] the authors introduced inhomogeneous bases of the ring of symmetric functions. The elements in these bases have the property that they evaluate to characters of symmetric groups. In this article we develop further properties of these bases by proving product and coproduct formulae. In addition, we give the transition coefficients between the elementary symmetric functions and the irreducible character basis.

Highlights

In [23], the authors introduced a basis of the symmetric functions {sλ} that specialize to the characters of the irreducible modules of the symmetric group when the symmetric group Sn is embedded in GLn as permutation matrices
The Hopf algebra structure that we develop here lays out the combinatorics and indicates that a combinatorial interpretation for the stable Kronecker and restriction problems may exist using operations on multiset tableaux
In [23] we provided a combinatorial interpretation for the transition coefficients from the complete homogeneous basis to the sλ-basis in terms of multiset tableaux

Summary

Introduction

In [23], the authors introduced a basis of the symmetric functions {sλ} that specialize to the characters of the irreducible modules of the symmetric group when the symmetric group Sn is embedded in GLn as permutation matrices. Several outstanding open problems in combinatorial representation theory are encoded in the linear algebra related to this basis One such problem is the restriction problem [7, 15, 17, 22, 27, 28]. The Hopf algebra structure that we develop here lays out the combinatorics and indicates that a combinatorial interpretation for the stable Kronecker and restriction problems may exist using operations on multiset tableaux. In [23] we provided a combinatorial interpretation for the transition coefficients from the complete homogeneous basis to the sλ-basis in terms of multiset tableaux In representation theory, this is equivalent to computing the multiplicities when we restrict the tensor product of symmetric tensors from GLn to Sn. In this paper, we give a combinatorial interpretation for the expansion of the elementary basis in the irreducible character basis.

Notation and Preliminaries

Products of character bases

Findings

Coproducts of the character bases