TY - JOUR
T1 - q-dimensions of highest weight crystals and cyclic sieving phenomenon
AU - Oh, Young Tak
AU - Park, Euiyong
N1 - Publisher Copyright:
© 2021 Elsevier Ltd
PY - 2021/10
Y1 - 2021/10
N2 - 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(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.
AB - 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(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.
UR - http://www.scopus.com/inward/record.url?scp=85107738451&partnerID=8YFLogxK
U2 - 10.1016/j.ejc.2021.103372
DO - 10.1016/j.ejc.2021.103372
M3 - Article
AN - SCOPUS:85107738451
SN - 0195-6698
VL - 97
JO - European Journal of Combinatorics
JF - European Journal of Combinatorics
M1 - 103372
ER -