\(q\)-dimensions of highest weight crystals and cyclic sieving phenomenon. (English) Zbl 1470.05169
Summary: 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 P. Alexandersson and N. Amini [Discrete Math. 342, No. 6, 1581–1601 (2019; Zbl 1414.05300)]. 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 \((\mathsf{SST}_m (\lambda),\langle\mathsf{c}\rangle,\mathsf{s}_\lambda(1,q,q^2,\dots,q^{m-1}))\) 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 \(n\).
MSC:
| 05E10 | Combinatorial aspects of representation theory |
| 05E16 | Combinatorial aspects of groups and algebras |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 13A50 | Actions of groups on commutative rings; invariant theory |
Citations:
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