Orbit harmonics for the union of two orbits

TL;DR

This paper extends the concept of orbit harmonics to the union of two symmetric group orbits, showing how their combined structure relates to Springer representations and degree shifts.

Contribution

It introduces a novel analysis of orbit harmonics for unions of two orbits, revealing their decomposition into shifted Springer representations.

Findings

When coordinate sums differ, the union's graded representation splits into two Springer representations.

One of the Springer representations is shifted in degree by 1.

The work generalizes previous orbit harmonic constructions to unions of orbits.

Abstract

Garsia and Procesi, in their study of Springer's representation, proved that the cohomology ring of a Springer fiber is isomorphic to the associated graded ring of the coordinate ring of the $S_n$ orbit of a single point in $\mathbb{C}^n$. This construction was an essential tool in their analysis of the Springer representation, and variations of it have reappeared recently in several other combinatorial and geometric contexts under the name orbit harmonics. In this article, we analyze the orbit harmonics of a union of two $S_n$ orbits. We prove that when the coordinate sums of the two orbits are different, the corresponding graded $S_n$ representation is a direct sum of two Springer representations, one of which is shifted in degree by 1.