On fusion product and N!/k-conjecture

Abstract

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.