与 Macdonald 理论 delta 算子相关的旗簇推广

Betti numbers of smooth Schubert varieties and the remarkable formula of Kostant,Macdonald,Shapiro and Steinberg

The purpose of this note is to give a refinement of the product formula proved in [1] for the Poincare polynomial of a smooth Schubert variety in the flag variety of an algebraic group G over C. This yields a factorization of the number of elements in a Bruhat interval [e,w] in the Weyl group W of G provided the Schubert variety associated to w is smooth. This gives an elementary necessary condition for a Schubert variety in the flag variety to be smooth.

稳定极限部分对称Macdonald函数与抛物线旗Hilbert概形

Stable-limit partially symmetric Macdonald functions and parabolic flag Hilbert schemes

本文显式计算了抛物线旗Hilbert概形正规化不动点类在同构映射下的像，证明其与修正的部分对称Macdonald多项式一致，从而证实了先前猜想，并给出了对合作用的显式公式。

置换基 MacDonald 多项式恒等式的组合证明

Combinatorial proof of a permuted basement Macdonald polynomial identity

研究置换基 MacDonald 多项式恒等式，通过将 Concha - Lapointe 恒等式细化到特定子族，给出组合证明，并揭示其与 Kazhdan - Lusztig 对合的等价关系。

A flag variety for the Delta Conjecture

The Delta Conjecture of Haglund, Remmel, and Wilson predicts the monomial expansion of the symmetric function $Δ'_{e_{k-1}} e_n$, where $k \leq n$ are positive integers and $Δ'_{e_{k-1}}$ is a Macdonald eigenoperator. When $k = n$, the specialization $Δ'_{e_{n-1}} e_n|_{t = 0}$ is the Frobenius image of the graded $S_n$-module afforded by the cohomology ring of the {\em flag variety} consisting of complete flags in $\mathbb{C}^n$. We define and study a variety $X_{n,k}$ which carries an action of $S_n$ whose cohomology ring $H^{\bullet}(X_{n,k})$ has Frobenius image given by $Δ'_{e_{k-1}} e_n|…