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On the smooth locus of affine Schubert varieties

Georgios Pappas, Rong Zhou

Abstract

We give a simple and uniform proof of a conjecture of Haines-Richarz characterizing the smooth locus of Schubert varieties in twisted affine Grassmannians. Our method is elementary and avoids any representation theoretic techniques, instead relying on a combinatorial analysis of tangent spaces of Schubert varieties.

On the smooth locus of affine Schubert varieties

Abstract

We give a simple and uniform proof of a conjecture of Haines-Richarz characterizing the smooth locus of Schubert varieties in twisted affine Grassmannians. Our method is elementary and avoids any representation theoretic techniques, instead relying on a combinatorial analysis of tangent spaces of Schubert varieties.
Paper Structure (5 sections, 10 theorems, 65 equations)

This paper contains 5 sections, 10 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Tangent spaces of partial affine flag varieties
  3. Root curves in twisted affine Grassmannians
  4. Tangent spaces for minimal degenerations
  5. Example: The ramified triality

Key Result

Theorem 1.1

Let $\mathfrak{s}$ be an absolutely special vertex. Then the smooth locus of $\mathrm{Gr}_{\mathcal{G},\preccurlyeq\mu}$ is $\mathrm{Gr}_{\mathcal{G},\mu}$.

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.2.1
  • Remark 2.2.2
  • Proposition 2.5
  • proof
  • Proposition 2.7
  • proof
  • Proposition 2.9
  • proof
  • Proposition 3.4
  • ...and 10 more