Submatrices of character tables and basic sets

TL;DR

This paper explores the structure of submatrices of character tables in finite groups, revealing novel combinatorial and arithmetic properties, especially for symmetric groups and related symmetric functions.

Contribution

It uncovers unexpected combinatorial determinant formulas and arithmetic properties of character submatrices and transition matrices in symmetric groups and symmetric functions.

Findings

Determinant formulas for submatrices of character tables

Arithmetic properties of related combinatorial numbers

Connections between character tables and symmetric functions

Abstract

In this investigation of character tables of finite groups we study basic sets and associated representation theoretic data for complementary sets of conjugacy classes. For the symmetric groups we find unexpected properties of characters on restricted sets of conjugacy classes, like beautiful combinatorial determinant formulae for submatrices of the character table and Cartan matrices with respect to basic sets; we observe that similar phenomena occur for the transition matrices between power sum symmetric functions to bounded partitions and the $k$-Schur functions introduced by Lapointe and Morse. Arithmetic properties of the numbers occurring in this context are studied via generating functions.