$q$-dimensions of highest weight crystals and cyclic sieving phenomenon

Abstract

In this paper, we compute explicitly the $q$-dimensions of highest weight crystals modulo $q^n-1$ for a quantum group of arbitrary finite type under certain assumption, and interpret the modulo computations in terms of the cyclic sieving phenomenon. This interpretation gives an affirmative answer to the conjecture by Alexandersson and Amini. As an application, under the assumption that $\lambda$ is a partition of length $<m$ and there exists a fixed point in ${\mathsf{SST}}_m(\lambda )$ under the action $\mathsf{c}$ arising from the crystal structure, we show that the triple $({\mathsf{SST}}_m(\lambda ),⟨\mathsf{c}⟩,{\mathsf{s}}_{\lambda }(1,q,q^2,\dots ,q^{m-1}))$ exhibits the cycle sieving phenomenon if and only if $\lambda$ is of the form $((am{)}^b)$, where either $b=1$ or $m-1$. Moreover, in this case, we give an explicit formula to compute the number of all orbits of size $d$ for each divisor $d$ of $n$.

19 pages, minor revision, Lemma 2.3 changed; to appear in European Journal of Combinatorics

References in corpus (4)

• Skew characters and cyclic sieving
• The Cone of Cyclic Sieving Phenomena
• Restriction of Global Bases and Rhoades's Theorem
• Promotion and cyclic sieving on families of SSYT

Cited by in corpus (2)

• Skew characters and cyclic sieving
• Refined Catalan and Narayana cyclic sieving