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Unexpected toric Richardson varieties

Eugene Gorsky, Soyeon Kim, Melissa Sherman-Bennett

Abstract

We prove that an open Richardson variety in the complete flag variety for $\mathrm{GL}_n$ is isomorphic to a torus if and only if the corresponding closed Richardson variety is toric. Such toric varieties can be classified in terms of the combinatorics of Bruhat intervals, and include many varieties of dimension larger than $n-1$. We give a combinatorial description of the corresponding polytopes, and compute several explicit examples.

Unexpected toric Richardson varieties

Abstract

We prove that an open Richardson variety in the complete flag variety for is isomorphic to a torus if and only if the corresponding closed Richardson variety is toric. Such toric varieties can be classified in terms of the combinatorics of Bruhat intervals, and include many varieties of dimension larger than . We give a combinatorial description of the corresponding polytopes, and compute several explicit examples.

Paper Structure

This paper contains 20 sections, 53 theorems, 102 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Algebraic tori
  4. Richardson varieties
  5. Positroids and positroid varieties
  6. Deodhar decomposition and Marsh-Rietsch parametrization
  7. Combinatorics of Bruhat intervals
  8. A lemma on posets
  9. Toric Richardson varieties
  10. On open Richardsons which are tori
  11. On closed Richardsons which are toric
  12. Positroid projections
  13. Moment polytopes of toric Richardsons
  14. The moment polytopes as Minkowski sums
  15. Vertices and edges
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose the open Richardson variety $R_{v,w}$ is isomorphic to a torus $\mathbb{T}$. Then the action of $\mathbb{T}$ on $R_{v,w}$ extends to the closed Richardson variety $\overline{R}_{v,w}$ and $\overline{R}_{v,w}$ is a toric variety with respect to $\mathbb{T}$.

Figures (7)

  • Figure 1: The graph $G_{v, \mathbf{w}}$ for $v= 1324$ and $\mathbf{w}=s_1s_2s_3s_2s_1$
  • Figure 2: NI path collections on a 3D plabic graph and their weights.
  • Figure 3: The polytope $\widehat{P}(v,w)$ from $4$-crown
  • Figure 4: 3D plabic graph for $n=6$ for the family of examples in Section \ref{['subsec-infinite-family']}. The colored paths give examples of the types of paths in the proof of Lemma \ref{['lem: infinite series MR minor']} for $\Delta_{[4] \setminus 2 \cup 6}$: the type 1 path is in red, type 2 paths are in green, and the type 3 path is in blue.
  • Figure 5: For $n$ even, set $w=(1~n)$ and $v=s_2 s_4 \dots s_{n-2}$. The plabic graphs $G_k(v,w)$ for $k$ odd (left) and $k$ even (right).
  • ...and 2 more figures

Theorems & Definitions (116)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • proof
  • Definition 2.3
  • ...and 106 more