Grassmannian上二面角作用量的量子模拟
A quantum analogue of the dihedral action on Grassmannians
摘要：
在最近的工作中，Launois和Lenagan已经展示了如何构造量子格拉斯曼的上循环扭曲以及扭曲和非扭曲代数的同构，该同构将给定的量子子代数发送到其索引集根据$n$ -循环$c=(1, 2, ..., n)$排列的子代数，直到$q$的幂。这种扭曲是必要的，因为$c$不会像经典和半经典那样自然地诱导量子格拉斯曼的自同构。我们扩展了这个结构，给出了一个量子模拟的作用在格拉斯曼的二面体子群的$S_{n}$的$c$和$w_{0}$，最长的元素，而这种模拟的形式的广群。我们证明了该子群对量子Grassmannian的环面不变素理想的诱导作用，并证明了该子群对完全非负和完全正Grassmannian的诱导作用.然后我们看到，这种二面角子群作用经典地、半经典地（通过泊松自同构和反自同构，这是雅基莫夫的一个结果）存在于量子和非负环境中。
英文摘要：
In recent work, Launois and Lenagan have shown how to construct a cocycle twisting of the quantum Grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the $n$-cycle $c=(1, 2, ..., n)$, up to a power of $q$. This twisting is needed because $c$ does not naturally induce an automorphism of the quantum Grassmannian, as it does classically and semi-classically. We extend this construction to give a quantum analogue of the action on the Grassmannian of the dihedral subgroup of $S_{n}$ generated by $c$ and $w_{0}$, the longest element, and this analogue takes the form of a groupoid. We show that there is an induced action of this subgroup on the torus-invariant prime ideals of the quantum Grassmannian and also show that this subgroup acts on the totally nonnegative and totally positive Grassmannians. Then we see that this dihedral subgroup action exists classically, semi-classically (by Poisson automorphisms and anti-automorphisms, a result of Yakimov) and in the quantum and nonnegative settings.
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