论文标题
简单类型的标志未成年人的倍增性
Equivariant multiplicities of simply-laced type flag minors
论文作者
论文摘要
令$ \ mathfrak {g} $为有限的简单类型的简单谎言代数。 Baumann-Kamnitzer-Knutson最近在坐标环$ \ Mathbb {C} [c} [n] $上定义了与Brion的Equivariant Multiplicities通过几何表达相关的代数形态{D} $。已知该地图对与光滑MV循环相对应的MV基础的元素以及对应于Kleshchev-Ram在Quiver Hecke代数上强的均匀模块的元素进行区分值。在本文中，我们表明，当$ \ mathfrak {g} $是类型$ a_n $或$ d_4 $的类型时，地图$ \ overline {d} $在所有$ \ mathbb {c} [c} [c} [c} [c} [n] $的所有标志未成年人的集合上采用了类似的杰出值，从而提高了相应的MV Cycles的平滑性问题。我们还表现出属于同一标准种子的Flag未成年人的$ \叠加{d} $的值之间的某些关系，我们表明，在任何$ ade $ type中，这些关系都保留在一个从一个标准种子到另一个标准种子的群集突变下。这些结果的证据部分依赖于Kang-Kashiwara-kim-OH对$ \ mathbb {c} [c} [n] $的群集结构的单体分类，这是通过Quiver Hecke代数的表示。
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.
