The Combinatorics of the Garsia-Haiman Modules for Hook Shapes

A kicking basis for the two-column Garsia-Haiman modules

In the early 1990s, Garsia and Haiman conjectured that the dimension of the Garsia-Haiman module is n!, and they showed that the resolution of this conjecture implies the Macdonald Positivity Conjecture. Haiman proved these conjectures in 2001 using algebraic geometry, but the question remains to find an explicit basis for the module which would give a simple proof of the dimension. Using the theory of Orbit Harmonics developed by Garsia and Haiman, we present a "kicking basis" for Garsia-Haiman modules indexed by a partition with at most two columns.

A proof of the $\frac{n!}{2}$ conjecture for hook shapes

A well-known representation-theoretic model for the transformed Macdonald polynomial $\widetilde{H}_μ(Z;t,q)$, where $μ$ is an integer partition, is given by the Garsia-Haiman module $\mathcal{H}_μ$. We study the $\frac{n!}{k}$ conjecture of Bergeron and Garsia, which concerns the behavior of certain $k$-tuples of Garsia-Haiman modules under intersection. In the special case that $μ$ has hook shape, we use a basis for $\mathcal{H}_μ$ due to Adin, Remmel, and Roichman to resolve the $\frac{n!}{2}$ conjecture by constructing an explicit basis for the intersection of two Garsia-Haiman modules.

关于融合积与 N!/k-猜想

On fusion product and N!/k-conjecture

本文为当前李代数建立 Schur--Weyl 对偶，将 Garsia--Haiman 模表示为融合积，证明 Butler 猜想并给出 N!/k 猜想维数下界。

Plethystic identities and mixed Hodge structures of character varieties

I demonstrate how certain identities for Macdonald's polynomials established by Garsia, Haiman and Tesler, together with the conjecture of Hausel, Letellier and Villegas imply explicit relations between mixed Hodge polynomials of different character varieties.

Garsia-Procesi 模块的一个下降基

A descent basis for the Garsia-Procesi module

本文为 Garsia-Procesi 模块 $R_λ$ 构造了一个基于 Garsia-Stanton 下降单项式的下降基，并通过 Springer 作用下的反对称化得到了抛物情形的基与多项式展开公式。

Kronecker coefficients for one hook shape

We give a positive combinatorial formula for the Kronecker coefficient g_{lambda mu(d) nu} for any partitions lambda, nu of n and hook shape mu(d) := (n-d,1^d). Our main tool is Haiman's \emph{mixed insertion}. This is a generalization of Schensted insertion to \emph{colored words}, words in the alphabet of barred letters \bar{1},\bar{2},... and unbarred letters 1,2,.... We define the set of \emph{colored Yamanouchi tableaux of content lambda and total color d} (CYT_{lambda, d}) to be the set of mixed insertion tableaux of colored words w with exactly d barred letters and such that w^{blft} is…