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Complexity of the Zero Set of a Matrix Schubert Ideal

Laura Escobar, Cesar Meza

TL;DR

This work classifies the possible complexities of the Y_w factors arising from matrix Schubert varieties under a torus action. By exploiting Rothe diagram data, L and L' diagrams, and an associated bipartite graph G^w, the authors derive a precise complexity formula d_w=|L'(w)|-|V(G^w)|+|C(G^w)| and prove that the maximal complexity for n\\ge 4 is d_{max}(n)=(n-1)(n-3), uniquely attained by w= w_0 s_{n-1}=[n,n-1,...,3,1,2]. Furthermore, they show that every integer d in {0,2,3,...,(n-1)(n-3)} occurs as d_w for some w\\in S_n, establishing a full range of complexities aside from 1. The results use a blend of combinatorial diagrammatics and polyhedral (weight cone) techniques to connect geometric complexity with diagrammatic and graph-theoretic invariants, providing a detailed map of how the torus action constrains the zero-set structure of matrix Schubert ideals.

Abstract

$T$-varieties are normal varieties equipped with an action of an algebraic torus $T$. When the action is effective, the complexity of a $T$-variety $X$ is $\dim(X)-\dim(T)$. Matrix Schubert varieties, introduced by Fulton in 1992, are $T$-varieties consisting of $n \times n$ matrices satisfying certain constraints on the ranks of their submatrices. In this paper, we focus on the complexity of certain torus-fixed affine subvarieties of matrix Schubert varieties. Concretely, given a matrix Schubert variety $\overline{X_{w}}$ where $w\in S_n$, we study the complexity of $Y_w$ obtained by the decomposition $\overline{X_{w}} = Y_{w} \times \mathbb{C}^{k}$ with $k$ as large as possible. Building up from results by Escobar and Mészáros and Donten-Bury, Escobar, and Portakal, we show that for a fixed $n$, the complexity of $Y_{w}$ with respect to this action can be any integer between $0$ and $(n-1)(n-3)$, except $1$.

Complexity of the Zero Set of a Matrix Schubert Ideal

TL;DR

This work classifies the possible complexities of the Y_w factors arising from matrix Schubert varieties under a torus action. By exploiting Rothe diagram data, L and L' diagrams, and an associated bipartite graph G^w, the authors derive a precise complexity formula d_w=|L'(w)|-|V(G^w)|+|C(G^w)| and prove that the maximal complexity for n\\ge 4 is d_{max}(n)=(n-1)(n-3), uniquely attained by w= w_0 s_{n-1}=[n,n-1,...,3,1,2]. Furthermore, they show that every integer d in {0,2,3,...,(n-1)(n-3)} occurs as d_w for some w\\in S_n, establishing a full range of complexities aside from 1. The results use a blend of combinatorial diagrammatics and polyhedral (weight cone) techniques to connect geometric complexity with diagrammatic and graph-theoretic invariants, providing a detailed map of how the torus action constrains the zero-set structure of matrix Schubert ideals.

Abstract

-varieties are normal varieties equipped with an action of an algebraic torus . When the action is effective, the complexity of a -variety is . Matrix Schubert varieties, introduced by Fulton in 1992, are -varieties consisting of matrices satisfying certain constraints on the ranks of their submatrices. In this paper, we focus on the complexity of certain torus-fixed affine subvarieties of matrix Schubert varieties. Concretely, given a matrix Schubert variety where , we study the complexity of obtained by the decomposition with as large as possible. Building up from results by Escobar and Mészáros and Donten-Bury, Escobar, and Portakal, we show that for a fixed , the complexity of with respect to this action can be any integer between and , except .

Paper Structure

This paper contains 7 sections, 6 theorems, 34 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. T-varieties
  4. Matrix Schubert varieties
  5. The variety Yw and its complexity
  6. The torus action on Yw
  7. Main Results

Key Result

Theorem 1

Fix $n \geq 4$. With respect to the ${\mathsf{T}} \times {\mathsf{T}}$-action, the maximum over all $w \in S_{n}$ of the complexity of the $T$-variety $Y_{w}$ is $(n-1)(n-3)$. The unique permutation at which this maximum is achieved is $[n,n-1,n-2, \dotsc, 3,1,2]$. In addition, for any $d \in \{0,2,

Figures (8)

  • Figure 2.1: The submatrix $M_{\square}^{a, b}$ of $M$. This figure is adapted from Escobar:2016aa.
  • Figure 2.2: The opposite Rothe diagram of the permutation $34512$.
  • Figure 2.3: The opposite Rothe diagram of the permutation $3412$.
  • Figure 3.1: The opposite Rothe diagram, southwest diagram, $L$-diagram, and $L'$-diagram of the permutation $3412$ and the $L'$-diagram of the permutation $51423$.
  • Figure 3.2: The bipartite graph $G^{3412}$
  • ...and 3 more figures

Theorems & Definitions (15)

  • Theorem : \ref{['thm: max-complexity']}, \ref{['thm: all complexities']}
  • Theorem 2.7: Fulton:1992aa
  • Example 2.8
  • Remark 3.1
  • Example 3.3
  • Lemma 3.4: Donten-Bury:2023aa
  • Example 3.7
  • Example 3.9
  • Theorem 4.1
  • proof
  • ...and 5 more