Cyclic Sieving and Plethysm Coefficients

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 $⟨p_{n/d}^d\circ s_{\mu },s_{\lambda }⟩$ is shown to count, up to sign, the number of fixed points of an $⟨s_{\mu }^n,s_{\lambda }⟩$-element set under the $d^{\text{th}}$ power of an order-$n$ cyclic action. If $n=2$, the action is the Schützenberger involution on semistandard Young tableaux (also known as evacuation), and, if $\lambda$ is rectangular, the action is a certain power of Schützenberger and Shimozono's jeu-de-taquin promotion. This work extends results of Stembridge and Rhoades linking fixed points of the Schützenberger actions to ribbon tableaux enumeration. The conclusion for the case $n=2$ is equivalent to the domino tableaux rule of Carré and Leclerc for discriminating between the symmetric and antisymmetric parts of the square of a Schur function.

28 pages, to appear in Trans. Amer. Math. Soc

References in corpus (1)

• Cyclic Sieving and Plethysm Coefficients

Cited by in corpus (3)

• Plethysm and lattice point counting
• Restriction of Global Bases and Rhoades's Theorem
• Cyclic Sieving and Plethysm Coefficients