Invariant Tensors and the Cyclic Sieving Phenomenon
Invariant Tensors and the Cyclic Sieving Phenomenon
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不变张量和循环筛选现象
DOI：
10.37236/4569
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
Bruce W. Westbury
中科院分区：
文献类型：
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作者：
Bruce W. Westbury
利用半简单李代数的表示理论，构造了一大类循环筛分现象的实例。设$M$为半简单李代数的有限维表示，设$B$为相关的柏原晶体。对于$r\ge 0$，表现出循环筛分现象的三元$(X,c,P)$结构如下：集合$X$是晶体中孤立顶点的集合$\otimes^rB$；映射$c\colon X\rightarrow X$是作用于矩形形状的标准表的推广，多项式$P$是$\mathfrak{S}_r$表示的Frobenius特征的假度，该表示与$\mathfrak{S}_r$在$\otimes^rM$中不变张量的子空间上的自然作用有关。以$M$作为$\mathrm{SL}(n)$的定义表示，给出了矩形画面的循环筛分现象。
We construct a large class of examples of the cyclic sieving phenomenon by expoiting the representation theory of semi-simple Lie algebras. Let $M$ be a finite dimensional representation of a semi-simple Lie algebra and let $B$ be the associated Kashiwara crystal. For $r\ge 0$, the triple $(X,c,P)$ which exhibits the cyclic sieving phenomenon is constructed as follows: the set $X$ is the set of isolated vertices in the crystal $\otimes^rB$; the map $c\colon X\rightarrow X$ is a generalisation of promotion acting on standard tableaux of rectangular shape and the polynomial $P$ is the fake degree of the Frobenius character of a representation of $\mathfrak{S}_r$ related to the natural action of $\mathfrak{S}_r$ on the subspace of invariant tensors in $\otimes^rM$. Taking $M$ to be the defining representation of $\mathrm{SL}(n)$ gives the cyclic sieving phenomenon for rectangular tableaux.