论文标题
$ Q $ - 重量最高的晶体和环状筛分现象
$q$-dimensions of highest weight crystals and cyclic sieving phenomenon
论文作者
论文摘要
在本文中，我们明确计算了最高重量晶体的$ q $ dimensions modulo $ q^n-1 $，以在某些假设下进行任意有限类型的量子组，并根据环状筛分现象来解释模型计算。这种解释对亚历山大和阿米尼的猜想提供了肯定的答案。作为一个应用，在假设$λ$是长度$ <m $的分区的假设下，并且在$ \ mathsf {c} $下，$ \ mathsf {sst} _m（λ）$中存在一个固定点\ rangle，\ mathsf {s}_λ（1，q，q^2，\ ldots，q^{m-1}））$在且仅当$λ$属于$ $（（am）^{b}）$的情况下，显示循环筛分现象。此外，在这种情况下，我们给出一个明确的公式，以计算每个除数$ d $ $ n $的尺寸$ d $的所有轨道的数量。
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 $λ$ is a partition of length $<m$ and there exists a fixed point in $\mathsf{SST}_m(λ)$ under the action $\mathsf{c}$ arising from the crystal structure, we show that the triple $(\mathsf{SST}_m(λ), \langle \mathsf{c} \rangle, \mathsf{s}_λ(1,q,q^2, \ldots, q^{m-1}))$ 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 $n$.
