A geometric realization of Catalan functions

TL;DR

This paper geometrically constructs a variety related to Catalan functions, proving key conjectures in algebraic combinatorics through geometric analysis.

Contribution

It introduces a new geometric model for Catalan functions and proves several longstanding conjectures in the field.

Findings

Proved the vanishing conjecture of Chen--Haiman.

Confirmed the tame case of the vanishing conjecture of Blasiak--Morse--Pun.

Established the monotonicity conjectures of Shimozono--Weyman.

Abstract

We construct a smooth projective variety $\mathscr X_\Psi$ that compactifies an equivariant vector subbundle of the cotangent bundle of the flag variety for $\mathrm{GL}(n)$, determined by a root ideal $\Psi$. A natural family of line bundles on $\mathscr X_\Psi$ yields the Catalan functions -- symmetric functions introduced by Chen--Haiman and studied further by Blasiak--Morse--Pun--Summers. By analyzing the geometry of $\mathscr X_\Psi$, we prove the vanishing conjecture of Chen--Haiman, confirm the tame case of the vanishing conjecture of Blasiak--Morse--Pun, and establish the monotonicity conjectures of Shimozono--Weyman.