arXiv 2005.070512020-05math.RT

简单连型旗子式的等变重数

Equivariant multiplicities of simply-laced type flag minors

Elie Casbi

备注：: Version 3, to appear in Representation Theory

摘要：

设$\mathfrak{g}$是有限单花边型单李代数. Baumann-Kamnitzer-Knutson最近通过几何Satake对应在与Brion等变重数相关的坐标环$\mathbb{C}[N]$上定义了一个代数态射$\overline{D}$。已知该映射在对应于光滑MV圈的MV基的元素上取有区别的值，以及在对应于Kleshchev-Ram强齐次模的对偶正则基的元素上取有区别的值。在本文中，我们表明，当$\mathfrak{g}$是类型$A_n$或$D_4$，映射$\overline{D}$采取类似的区别值的集合上的所有标志子的$\mathbb{C}[N]$，提出了相应的MV周期的光滑性的问题。我们还表现出一定的关系，$\overline{D}$的值属于同一标准种子的标志未成年人，我们表明，在任何$ADE$类型这些关系下保存从一个标准种子到另一个集群突变。这些结果的证明部分地依赖于Kang-Kashiwara-Kim-Oh对$\mathbb{C}[N]$的簇结构通过Hecke代数的表示进行的Monoidal分类。

英文摘要：

Let $\mathfrak{g}$ be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism $\overline{D}$ on the coordinate ring $\mathbb{C}[N]$ related to Brion's equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram's strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when $\mathfrak{g}$ is of type $A_n$ or $D_4$, the map $\overline{D}$ takes similar distinguished values on the set of all flag minors of $\mathbb{C}[N]$, raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of $\overline{D}$ on flag minors belonging to the same standard seed, and we show that in any $ADE$ type these relations are preserved under cluster mutations from one standard seed to another. The proofs of theses results partly rely on Kang-Kashiwara-Kim-Oh's monoidal categorification of the cluster structure of $\mathbb{C}[N]$ via representations of quiver Hecke algebras.

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