Orbit harmonics for the union of two 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.