Restriction of Global Bases and Rhoades's Theorem

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.

arXiv comment17 pages

References in corpus (4)

• Cyclic Sieving and Cluster Duality of Grassmannian
• Cyclic Sieving for Plane Partitions and Symmetry
• Cyclic Sieving and Plethysm Coefficients
• Cyclic sieving, rotation, and geometric representation theory

Cited by in corpus (3)

• Cyclic Sieving for Plane Partitions and Symmetry
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