Presenting the cohomology of a Schubert variety: Proof of the minimality conjecture

TL;DR

This paper proves a conjecture about the minimal number of generators needed for the cohomology rings of Schubert varieties, using symmetric function Hopf algebra structures, establishing fundamental lower bounds.

Contribution

It provides a proof of the minimality conjecture, confirming the exponential lower bound on generators for cohomology rings of Schubert varieties.

Findings

Proves the minimality conjecture for Schubert variety cohomology presentations.

Establishes an exponential lower bound on the number of generators.

Uses Hopf algebra structure of symmetric functions in the proof.

Abstract

A minimal presentation of the cohomology ring of the flag manifold $GL_n/B$ was given in [A. Borel, 1953]. This presentation was extended by [E. Akyildiz-A. Lascoux-P. Pragacz, 1992] to a non-minimal one for all Schubert varieties. Work of [Gasharov-Reiner, 2002] gave a short, i.e. polynomial-size, presentation for a subclass of Schubert varieties that includes the smooth ones. In [V. Reiner-A. Woo-A. Yong, 2011], a general shortening was found; it implies an exponential upper bound of $2^n$ on the number of generators required. That work states a minimality conjecture whose significance would be an exponential lower bound of $\sqrt{2}^{n+2}/\sqrt{\pi n}$ on the number of generators needed in worst case, giving the first obstructions to short presentations. We prove the minimality conjecture. Our proof uses the Hopf algebra structure of the ring of symmetric functions.