Symmetric group characters as symmetric functions

Abstract

We introduce a basis of the symmetric functions that evaluates to the (irreducible) characters of the symmetric group, just as the Schur functions evaluate to the irreducible characters of $G{L}_n$ modules. Our main result gives three different characterizations for this basis. One of the characterizations shows that the structure coefficients for the (outer) product of these functions are the stable Kronecker coefficients. The results in this paper focus on developing the fundamental properties of this basis.

arXiv comment35 pages; this is the more complete version of arXiv:1510.00438; v5 differs from previous versions with minor edits and the addition of an appendix about using Sagemath to do computations with these bases (this appendix does not appear in the published journal version)

References in corpus (6)

• FI-modules and stability for representations of symmetric groups
• Representation theory and homological stability
• The stability of the Kronecker products of Schur functions
• Reduced Kronecker coefficients and counter-examples to Mulmuley's strong saturation conjecture SH
• Specht modules decompose as alternating sums of restrictions of Schur modules
• Reduced Kronecker products which are multiplicity free or contain only few components

Cited by in corpus (5)

• Spaces of states of the two-dimensional O(n) and Potts models
• Moment curves and cyclic symmetry for positive Grassmannians
• Polynomial Induction and the Restriction Problem
• All Kronecker coefficients are reduced Kronecker coefficients
• A new approach to the representation theory of the partition category