The residue-table meta-conjecture for k-Schur functions

Let $k$ be a positive integer, let $u$ be a $k$-bounded partition, and let $R( u)$ be its residue table. The theory also involves $k$-bounded partitions and $k$-Schur functions, including operations such as $k$-conjugation, weak horizontal and vertical strips, and the Murnaghan–Nakayama rule. Residue-table meta-conjecture. Almost everything in the theory of $k$-bounded partitions and $k$-Schur functions can be expressed in an elegant way in terms of residue tables. This is an explicitly described, admittedly vague, guiding statement; the paper provides supporting evidence through several results and conjectural descriptions.