Quantum analogues of Schubert varieties in the grassmannian

Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases

We exhibit quantum cluster algebra structures on quantum Grassmannians $K_q[Gr(2,n)]$ and their quantum Schubert cells, as well as on $K_q[Gr(3,6)]$, $K_q[Gr(3,7)]$ and $K_q[Gr(3,8)]$. These cases are precisely those where the quantum cluster algebra is of finite type and the structures we describe quantize those found by Scott for the classical situation.

Prime ideals in the quantum grassmannian

We consider quantum Schubert cells in the quantum grassmannian and give a cell decomposition of the prime spectrum via the Schubert cells. As a consequence, we show that all primes are completely prime in the generic case where the deformation parameter q is not a root of unity. There is a torus H that acts naturally on the quantum grassmannian and the cell decomposition of the set of H-primes leads to a parameterisation of the H-spectrum via certain diagrams on partitions associated to the Schubert cells. Interestingly, the same parameterisation occurs for the non-negative cells in recent stu…

Reducedness of affine Grassmannian slices in type A

We prove in type A a conjecture which describes the ideal of transversal slices to spherical Schubert varieties in the affine Grassmannian. As a corollary, we prove a modular description (due to Finkelberg-Mirković) of the spherical Schubert varieties.

单位根处的量子格拉斯曼流形及其相关的量子舒伯特簇

Quantum Grassmannians and their Associated Quantum Schubert Varieties at roots of unity

本文研究了单位根处量子代数的 PI degree，通过证明相关矩阵不变因子为 2 的幂来显式计算分划子代数的 PI degree，并利用非交换去齐次化方法推导出量子舒伯特簇和量子格拉斯曼流形的 PI degree。

Quantum toric degeneration of quantum flag and Schubert varieties

We show that certain homological regularity properties of graded connected algebras, such as being AS-Gorenstein or AS-Cohen-Macaulay, can be tested by passing to associated graded rings. In the spirit of noncommutative algebraic geometry, this can be seen as an analogue of the classical result that, in a flat family of varieties over the affine line, regularity properties of the exceptional fiber extend to all fibers. We then show that quantized coordinate rings of flag varieties and Schubert varieties can be filtered so that the associated graded rings are twisted semigroup rings. This is a …

Geometric Crystals on Schubert Varieties

We define geometric crystals and unipotent crystals for arbitrary Kac-Moody groups and describe geometric and unipotent crystal structures on the Schubert varieties.