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

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 prove (and, in some cases, 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.