The mystery of plethysm coefficients

Understand

Composing two representations of the general linear groups gives rise to Littlewood's (outer) plethysm.

• On the level of characters, this poses the question of finding the Schur expansion of the plethysm of two Schur functions.
• A combinatorial interpretation for the Schur expansion coefficients of the plethysm of two Schur functions is, in general, still an open problem.
• We identify a proof technique of combinatorial representation theory, which we call the "$s$-perp trick", and point out several examples in the literature where this idea is used.