q-dimensions of highest weight crystals and cyclic sieving phenomenon

Young Tak Oh, Euiyong Park

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In this paper, we compute explicitly the q-dimensions of highest weight crystals modulo qn−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 λ is a partition of length <m and there exists a fixed point in SSTm(λ) under the action c arising from the crystal structure, we show that the triple (SSTm(λ),〈c〉,sλ(1,q,q2,…,qm−1)) exhibits the cycle sieving phenomenon if and only if λ 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.

Original languageEnglish
Article number103372
JournalEuropean Journal of Combinatorics
StatePublished - Oct 2021


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