论文标题
通过杰出的cominuscule家族迈向Landau-Ginzburg模型
Towards Landau-Ginzburg models for cominuscule spaces via the exceptional cominuscule family
论文作者
论文摘要
我们为特殊的cominuscule同质空间$ \ MATHBB {op}^2 = E_6^\ Mathrm {sc}/p_6 $和$ e_7^\ mathrm {sc}/p_7 $，分别为Cayley Plane和FreeDEnthal vorter。这些模型是在“ langlands双均匀空间”的反典型分隔的补充上定义的。格拉斯曼尼亚人，Quadrics和Lagrangian Grassmannians in Arxiv：1307.1085，Arxiv：1404.4844，Arxiv：1304.4958。我们证明，这些针对特殊家庭的模型与Arxiv：Math/0511124在Arxiv：Math/0511124中定义的Lie理论镜像模型同构，该模型在Arxiv：1912.09122中所证明，使用了对代数圆环（也称为Lusztig torus）。我们还提供了$ \ mathbb {c} [\ Mathbb {x}^\ vee] $的群集结构，证明plücker坐标构成了使用lusztig torus定义的估值基础的基础，并计算与此估值相关的牛顿 - 科诺科夫身体。尽管我们介绍了针对特殊类型的方法，但它们立即将其推广到其他Cominuscule家族的成员。
We present projective Landau-Ginzburg models for the exceptional cominuscule homogeneous spaces $\mathbb{OP}^2 = E_6^\mathrm{sc}/P_6$ and $E_7^\mathrm{sc}/P_7$, known respectively as the Cayley plane and the Freudenthal variety. These models are defined on the complement $X^\vee_\mathrm{can}$ of an anti-canonical divisor of the "Langlands dual homogeneous spaces" $\mathbb{X}^\vee = P^\vee\backslash G^\vee$ in terms of generalized Plücker coordinates, analogous to the canonical models defined for Grassmannians, quadrics and Lagrangian Grassmannians in arXiv:1307.1085, arXiv:1404.4844, arXiv:1304.4958. We prove that these models for the exceptional family are isomorphic to the Lie-theoretic mirror models defined in arXiv:math/0511124 using a restriction to an algebraic torus, also known as the Lusztig torus, as proven in arXiv:1912.09122. We also give a cluster structure on $\mathbb{C}[\mathbb{X}^\vee]$, prove that the Plücker coordinates form a Khovanskii basis for a valuation defined using the Lusztig torus, and compute the Newton-Okounkov body associated to this valuation. Although we present our methods for the exceptional types, they generalize immediately to the members of other cominuscule families.
