Another proof of the $n!$ conjecture

TL;DR

This paper offers a new proof of the $n!$ conjecture for staircase partitions by constructing a specific family of subschemes, simplifying the proof process and reinforcing Haiman's theorem.

Contribution

It introduces a novel geometric construction to prove the $n!$ conjecture for staircase partitions, providing an alternative proof of Haiman's theorem.

Findings

Proved the $n!$ conjecture for staircase partitions.

Provided a new geometric approach to the conjecture.

Reinforced the validity of Haiman's theorem.

Abstract

The "n! conjecture" of Garsia and Haiman has inspired mathematicians for nearly two decades, even after Haiman published a proof in 2001. Kumar and Funch Thomsen proved in 2003 that in order to prove the conjecture for all partitions, it suffices to prove it for the so-called "staircase partitions" $(k,k-1,...,2,1)$ for each $k>1$. In the present paper we give a construction of a specially designed two-dimensional family of length-$n$ subschemes of the plane, and use that to prove the $n!$ conjecture for staircase partitions. Together with the result of Kumar and Funch Thomsen, this provides a new proof of Haiman's theorem.