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The Isomorphism Problem for cominuscule Schubert Varieties

Edward Richmond, Mihail Tarigradschi, Weihong Xu

Abstract

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent.

The Isomorphism Problem for cominuscule Schubert Varieties

Abstract

Cominuscule flag varieties generalize Grassmannians to other Lie types. Schubert varieties in cominuscule flag varieties are indexed by posets of roots labeled long/short. These labeled posets generalize Young diagrams. We prove that Schubert varieties in potentially different cominuscule flag varieties are isomorphic as varieties if and only if their corresponding labeled posets are isomorphic, generalizing the classification of Grassmannian Schubert varieties using Young diagrams by the last two authors. Our proof is type-independent.
Paper Structure (6 sections, 12 theorems, 31 equations, 2 figures, 2 tables)

This paper contains 6 sections, 12 theorems, 31 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Examples of \ref{['thm:main']}
  3. Preliminaries
  4. Proof of \ref{['thm:main']}
  5. Forward direction: the isomorphism class of $X_\lambda$ determines the labeled poset $\lambda$
  6. Converse direction: the labeled poset $\lambda$ determines the isomorphism class of $X_\lambda$

Key Result

Theorem 1

Let $X_\lambda\subseteq X$ and $Y_{\mu}\subseteq Y$ be cominuscule Schubert varieties indexed by lower order ideals $\lambda\subseteq{\mathcal{P}}_X$ and $\mu\subseteq{\mathcal{P}}_Y$, respectively. Then $X_\lambda$ and $Y_{\mu}$ are algebraically isomorphic if and only if $\lambda$ and $\mu$ are is

Figures (2)

  • Figure 1: The Bruhat poset $W^I$ when $X=\mathop{\mathrm{Gr}}\nolimits(2,4)$. Permutations in $W^I$ are denoted using one-line notation, and next to each is the corresponding lower order ideal in ${\mathcal{P}}_X$. Join-irreducible elements of $W^I$ are the ones underlined, and the generator of the corresponding principal lower order ideal in ${\mathcal{P}}_X$ is decorated with a $\star$.
  • Figure 2: We highlight ${\mathcal{P}}_X^\Delta\subset {\mathcal{P}}_X$ with a bold border and label its boxes by the images of $\delta$.

Theorems & Definitions (35)

  • Theorem 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Example 8
  • Example 9
  • Proposition 10: Thomas--Yong, Buch--Samuel
  • ...and 25 more