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 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.