Restriction of Global Bases and Rhoades's Theorem

TL;DR

This paper proves a restriction property of global bases in quantum group representations and derives Rhoades's theorem on tableaux fixed under promotion using basis actions, linking it to Stembridge's evacuation result.

Contribution

It establishes a restriction property of global bases for certain quantum group representations and provides a new, concise proof of Rhoades's theorem on tableaux fixed under promotion.

Findings

Global bases decompose under restriction to smaller quantum groups.

Rhoades's action of the long cycle is derived from Berenstein--Zelevinsky's description.

A short proof of Rhoades's tableaux fixed under promotion is obtained.

Abstract

It is shown that if $\lambda$ is a multiple of a fundamental weight of $\mathfrak{sl}_k$, the lower global basis of the irreducible $U_q(\mathfrak{sl}_k)$-representation $V^{\lambda}$ with highest weight $\lambda$ comprises the disjoint union of the lower global bases of the irreducible $U_q(\mathfrak{sl}_{k-1})$-representations appearing in the decomposition of the restriction of $V^{\lambda}$ to $U_q(\mathfrak{sl}_{k-1})$. Rhoades's description of the action of the long cycle on the dual canonical basis of $V^{\lambda}$ is then deduced from Berenstein--Zelevinsky's description of the action of the long element. This yields a short proof of Rhoades's result on tableaux fixed under promotion which directly relates it to Stembridge's result on tableaux fixed under evacuation.