发表机构
University of Haifa; Yanqi Lake Beijing Institute of Mathematical Sciences And Applications (BIMSA)（海法大学; 北京雁栖湖应用数学研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
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。
AI 中文摘要
我们为当前李代数 $\mathfrak{gl}_V[x,y]=\mathfrak{gl}_V\otimes\mathbb{C}[x,y]$ 提出一个 Schur--Weyl 对偶的版本。在此对偶下，$N!$-猜想中的 Garsia--Haiman 模成为循环且余循环的 $\mathfrak{gl}_V[x,y]$-模，我们将其描述为平凡 $\mathfrak{gl}_V$-模的迭代融合积与余融合积。我们证明：对于某个图，Haiman 的 $N!$-定理等价于其行所对应的局部 Weyl 模的张量积上融合滤过与余融合滤过的一致性，并且 Garsia--Haiman 模的融合积与余融合积满足结合律。设 $\lambda$ 和 $\mu$ 是从同一图中移除两个不同角所得的 Young 图。我们构造一个 $S^2V$ 与局部 Weyl 模的迭代融合积，它是 $\lambda$ 和 $\mu$ 的 Garsia--Haiman 模的公共商的商，与它们各自构成短正合列，并且其特征为 Butler 的交多项式。这给出了 Butler 猜想的一个表示论证明。同样的构造给出了 $N!/k$-猜想中维数的下界。
英文摘要
We formulate a version of Schur--Weyl duality for the current Lie algebra $\mathfrak{gl}_V[x,y]=\mathfrak{gl}_V\otimes\mathbb{C}[x,y]$. Under this duality, the Garsia--Haiman modules of the $N!$-conjecture become cyclic and cocyclic $\mathfrak{gl}_V[x,y]$-modules, and we describe them as iterated fusion and cofusion products of tautological $\mathfrak{gl}_V$-modules. We show that Haiman's $N!$-theorem for a diagram is equivalent to the coincidence of the fusion and the cofusion filtrations on the tensor product of the local Weyl modules attached to its rows, and that the fusion and cofusion products of Garsia--Haiman modules are associative. Let $λ$ and $μ$ be Young diagrams obtained by removing two different corners from the same diagram. We construct an iterated fusion product of $S^2V$ with local Weyl modules, which is a quotient of the common quotient of the Garsia--Haiman modules of $λ$ and $μ$, fits into short exact sequences with each of them, and has Butler's intersection polynomial as its character. This gives a representation-theoretic proof of Butler's conjecture. The same construction gives lower bounds for the dimensions in the $N!/k$-conjecture.
Comments74 pages. First draft of a long-standing project on the relation between Feigin-Loktev fusion products and Haiman's N!-theorem. Includes a representation-theoretic proof of Butler's conjecture and lower bounds for the N!/k-conjecture. Comments are welcome
