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 ${\mathsf{SST}}_m\left(\lambda \right)$ under the action $\mathsf{c}$ arising from the crystal structure, we show that the triple $({\mathsf{SST}}_m\left(\lambda \right),〈\mathsf{c}〉,{\mathsf{s}}_{\lambda }(1,q,q^2,\dots ,q^{m-1}))$ exhibits the cycle sieving phenomenon if and only if $\lambda$ is of the form $\left({\left(am\right)}^b\right)$, 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 $n$.

在本文中，我们明确计算 $q$- 最高重量晶体模的尺寸 $q^n-1$对于某个假设下任意有限类型的量子群，并根据循环筛分现象解释模计算。这种解释对亚历山大松和阿米尼的猜想给出了肯定的回答。作为一个应用程序，假设$\lambda$ 是长度的划分 $<米$ 并且存在一个不动点 ${\mathsf{不锈钢}}_{米}\left(\lambda \right)$ 在行动之下 $\mathsf{C}$ 由晶体结构引起，我们表明三重 $({\mathsf{不锈钢}}_{米}\left(\lambda \right),〈\mathsf{C}〉,{\mathsf{秒}}_{\lambda }(1,q,q^2,\dots ,q^{米-1}))$ 表现出循环筛分现象当且仅当 $\lambda$ 是形式 $\left({\left(\mathrm{一种}米\right)}^{乙}\right)$，其中任一 $乙=1$ 或者 $米-1$. 此外，在这种情况下，我们给出了一个明确的公式来计算所有大小轨道的数量$d$ 对于每个除数 $d$ 的 $n$.