Cyclic Sieving and Plethysm Coefficients

TL;DR

This paper provides a combinatorial formula for plethysm coefficients in specific cases, linking fixed points of cyclic actions on tableaux to known combinatorial objects, extending previous results in algebraic combinatorics.

Contribution

It introduces a new combinatorial expression for plethysm coefficients when n=2 or λ is rectangular, connecting these coefficients to fixed points of cyclic actions on tableaux.

Findings

Provides a combinatorial expression for plethysm coefficients in special cases.

Shows these coefficients count fixed points of certain cyclic actions on tableaux.

Extends previous work relating tableau actions to ribbon and domino tableaux.

Abstract

A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In these cases, the coefficient $\langle p_{n/d}^d \circ s_{\mu}, s_{\lambda} \rangle$ is shown to count, up to sign, the number of fixed points of an $\langle s_{\mu}^n, s_{\lambda} \rangle$-element set under the $d^{\text{th}}$ power of an order-$n$ cyclic action. If $n=2$, the action is the Sch\"utzenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Sch\"utzenberger and Shimozono's jeu-de-taquin promotion. This work extends results of Stembridge and Rhoades linking fixed points of the Sch\"utzenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carr\'e and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.