A multiplicative property of quantum flag minors

A multiplicative property of quantum flag minors II

Let U^+ be the plus part of the quantized enveloping algebra of a simple Lie algebra and let B^* be the dual canonical basis of U^+. Let b,b' be in B* and suppose that one of the two elements is a q-commuting product of quantum flag minors. We show that b and b' are multiplicative if and only if they q-commute.

Equivariant multiplicities of simply-laced type flag minors

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 $…

Auslander代数、旗组合与量子旗簇

Auslander algebras, flag combinatorics and quantum flag varieties

本文通过构造Auslander代数D的good模子范畴F_Δ(J)并赋予其Frobenius稳定2-Calabi-Yau结构，实现了旗簇坐标环的加法范畴化，并证明了量子旗簇坐标环是量子簇代数。

量子群 $\dot U$ 与半域 Z 上的旗流形

The quantum group $\dot U$ and flag manifolds over the semifield Z

本文利用半域 Z 上旗流形，对修正量子群的典范基给出参数化，并修正了此前工作中的一些小错误。

A Chevalley formula for semi-infinite flag manifolds and quantum K-theory (Extended abstract)

We give a combinatorial Chevalley formula for an arbitrary weight, in the torus-equivariant K-theory of semi-infinite flag manifolds, which is expressed in terms of the quantum alcove model. As an application, we prove the Chevalley formula for anti-dominant fundamental weights in the (small) torus-equivariant quantum K-theory of the flag manifold G/B; this has been a longstanding conjecture about the multiplicative structure of the mentioned quantum K-theory. Moreover, in type A, we prove that the so-called quantum Grothendieck polynomials indeed represent Schubert classes in the (non-equivar…

Accessible Proof of Standard Monomial Basis for Coordinatization of Schubert Sets of Flags

The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. L…