量子K-理论除数公理的无消去版本：旗 manifold 在准次小情况下的应用

旗簇量子 $K$-理论中的微小关系

Minuscule Relations in Quantum $K$-Theory of Flag Varieties

本文研究旗簇的量子 $K$-理论，为每个微小基本权构造环面等变量子 $K$-理论中的显式关系，该关系是相应不可约表示特征的量子形变。

Inverse $K$-Chevalley formulas for semi-infinite flag manifolds, I: minuscule weights in ADE type

We prove an explicit inverse Chevalley formula in the equivariant $K$-theory of semi-infinite flag manifolds of simply-laced type. By an inverse Chevalley formula, we mean a formula for the product of an equivariant scalar with a Schubert class, expressed as a $\mathbb{Z}[q^{\pm 1}]$-linear combination of Schubert classes twisted by equivariant line bundles. Our formula applies to arbitrary Schubert classes in semi-infinite flag manifolds of simply-laced type and equivariant scalars $e^λ$, where $λ$ is an arbitrary minuscule weight. By a result of Stembridge, our formula completely determines …

Chevalley formula for anti-dominant minuscule fundamental weights in the equivariant quantum $K$-group of partial flag manifolds

In this paper, we give an explicit formula of Chevalley type, in terms of the Bruhat graph, for the quantum multiplication with the class of the line bundle associated to the anti-dominant minuscule fundamental weight $- \varpi_{k}$ in the torus-equivariant quantum $K$-group of the partial flag manifold $G/P_{J}$ (where $J = I \setminus \{k\}$) corresponding to the maximal (standard) parabolic subgroup $P_{J}$ of minuscule type in type $A$, $D$, $E$, or $B$. This result is obtained by proving a similar formula in a torus-equivariant $K$-group of the semi-infinite partial flag manifold $\mathbf…

The AS-Cohen-Macaulay property for quantum flag manifolds of minuscule weight

It is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights, their Schubert varieties, big cells, and determinantal varieties are AS-Cohen-Macaulay. The main ingredient in the proof is the notion of a quantum graded algebra with a straightening law, introduced by T.H. Lenagan and L. Rigal [J. Algebra 301 (2006), 670-702]. Using Stanley's Theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are AS-Gorenstein.

Solutions of the tt*-Toda Equations and Quantum Cohomology of Flag Manifolds

We relate the quantum cohomology of minuscule flag manifolds to the tt*-Toda equations, a special case of the topological-antitopological fusion equations which were introduced by Cecotti and Vafa in their study of supersymmetric quantum field theories. To do this, we combine the Lie-theoretic treatment of the tt*-Toda equations of Guest-Ho with the Lie-theoretic description of the quantum cohomology of minuscule flag manifolds from Chaput-Manivel-Perrin and Golyshev-Manivel.

旗流形的量子 K 理论除子公理

Quantum $K$-theoretic divisor axiom for flag manifolds

本文证明了旗流形 G/P 的（环面等变）3 点、亏格 0、K 理论 Gromov-Witten 不变量的恒等式，作为量子 K 理论中“除子公理”的替代，并利用 Chevalley 公式和量子 Bruhat 图实现了类型无关的证明与特定不变量的计算。