q-dimensions of highest weight crystals and cyclic sieving phenomenon

초록

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 SSTm(lambda) under the action c arising from the crystal structure, we show that the triple (SSTm(lambda), c , s(lambda)(1, q, q(2), ... , 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. (C) 2021 Elsevier Ltd. All rights reserved.

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