The role of residue and quotient tables in the theory of k-Schur functions

TL;DR

This paper explores how residue and quotient tables can be used to describe various aspects of the theory of k-Schur functions, potentially leading to a comprehensive rule for their products.

Contribution

It introduces the use of residue and quotient tables to unify and extend results in the theory of k-bounded partitions and k-Schur functions, including open problems.

Findings

Residue and quotient tables describe strong covers in k-bounded partitions.

They can potentially characterize k-conjugates and weak strips.

Evidence suggests they may help formulate a product rule for k-Schur functions.

Abstract

Recently, residue and quotient tables were defined by Fishel and the author, and were used to describe strong covers in the lattice of $k$-bounded partitions. In this paper, we show or conjecture that residue and quotient tables can be used to describe many other results in the theory of $k$-bounded partitions and $k$-Schur functions, including $k$-conjugates, weak horizontal and vertical strips, and the Murnaghan-Nakayama rule. Evidence is presented for the claim that one of the most important open questions in the theory of $k$-Schur functions, a general rule that would describe their product, can be also concisely stated in terms of residue tables.