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

Avery St. Dizier, Alexander Yong

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 $\sqrt{2}^{n+2}/\sqrt{πn}$ 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.

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

Abstract

A minimal presentation of the cohomology ring of the flag manifold 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 on the number of generators required. That work states a minimality conjecture whose significance would be an exponential lower bound of 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.
Paper Structure (7 sections, 15 theorems, 93 equations, 4 figures, 1 algorithm)

This paper contains 7 sections, 15 theorems, 93 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. The main result
  4. Organization
  5. Setting up the linear equations
  6. The Hopf Algebra of symmetric functions
  7. Conclusion of the proof

Key Result

Theorem 1.1

For any $w\in S_n$, the ideal $I_w$ defining $H^*(X_w)$ as a quotient of $H^*(X)$ is generated by the cohomology classes $\sigma_u$ where $u\not\leq w$ and $u$ is grassmannian.

Figures (4)

  • Figure 1:
  • Figure 2: The decomposable equations for Example \ref{['exp:equations']}, divided into tall and wide equations.
  • Figure 3: Example of Theorem \ref{['thm:pictures']}.
  • Figure 4: The Hopf algebra stucture on $\Lambda$

Theorems & Definitions (41)

  • Theorem 1.1: ALP
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • ...and 31 more