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The MatrixSchubert package for Macaulay2

Ayah Almousa, Sean Grate, Daoji Huang, Patricia Klein, Adam LaClair, Yuyuan Luo, Joseph McDonough

TL;DR

MatrixSchubert presents a Macaulay2 package for constructing and analyzing matrix Schubert varieties and ASM varieties, equipping researchers with tools to compute homological invariants, initial ideals, rank structures, and key polynomials. It leverages combinatorial data (Rothe diagrams, rank tables) and theoretical results (Knutson–Miller, Conca–Varbaro CV20, PSW22) to enable efficient computation of invariants such as Castelnuovo–Mumford regularity and to relate ASM ideals to intersections of Schubert determinantal ideals. The package provides implementations of Fulton generators, rank-table methods, initial-ideal machinery, and pattern-avoidance tests, along with diagonal term orders and pipe-dream related structures, to study codimension, Cohen–Macaulayness, and resolutions of matrix Schubert and ASM varieties. This work enhances computational access to open questions on resolutions, Betti numbers, CM properties, and component structure, facilitating both theoretical investigation and practical exploration across Schubert calculus and ASM combinatorics.

Abstract

We introduce the MatrixSchubert package for the computer algebra system Macaulay2. This package has tools to construct and study matrix Schubert varieties and alternating sign matrix (ASM) varieties. The package also introduces tools for quickly computing homological invariants of such varieties, finding the components of an ASM variety, and checking if a union of matrix Schubert varieties is an ASM variety.

The MatrixSchubert package for Macaulay2

TL;DR

MatrixSchubert presents a Macaulay2 package for constructing and analyzing matrix Schubert varieties and ASM varieties, equipping researchers with tools to compute homological invariants, initial ideals, rank structures, and key polynomials. It leverages combinatorial data (Rothe diagrams, rank tables) and theoretical results (Knutson–Miller, Conca–Varbaro CV20, PSW22) to enable efficient computation of invariants such as Castelnuovo–Mumford regularity and to relate ASM ideals to intersections of Schubert determinantal ideals. The package provides implementations of Fulton generators, rank-table methods, initial-ideal machinery, and pattern-avoidance tests, along with diagonal term orders and pipe-dream related structures, to study codimension, Cohen–Macaulayness, and resolutions of matrix Schubert and ASM varieties. This work enhances computational access to open questions on resolutions, Betti numbers, CM properties, and component structure, facilitating both theoretical investigation and practical exploration across Schubert calculus and ASM combinatorics.

Abstract

We introduce the MatrixSchubert package for the computer algebra system Macaulay2. This package has tools to construct and study matrix Schubert varieties and alternating sign matrix (ASM) varieties. The package also introduces tools for quickly computing homological invariants of such varieties, finding the components of an ASM variety, and checking if a union of matrix Schubert varieties is an ASM variety.
Paper Structure (14 sections, 4 theorems, 12 equations, 1 figure)

This paper contains 14 sections, 4 theorems, 12 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and some basic functions
  3. Permutations
  4. Alternating Sign Matrices
  5. Rank tables
  6. Pattern Avoidance
  7. Algorithms for Castelnuovo--Mumford regularity
  8. Matrix Schubert varieties
  9. ASM Varieties
  10. Schubert and Grothendieck polynomials
  11. Studying ASM varieties via initial ideals
  12. Some diagonal term orders
  13. Available Examples of ASMs
  14. Acknowledgements

Key Result

Proposition 2.1

For $w \in S_n$, $I_w$ is prime, $\hbox{codim}(I_w)=\ell(w)$, $R/I_w$ is Cohen--Macaulay.

Figures (1)

  • Figure 1: One pipe dream for $w = 214365$. + is interpreted as a cross tile, and / is interpreted as an elbow tile, and entries below the main antidiagonal are ignored, as is standard.

Theorems & Definitions (4)

  • Proposition 2.1: Ful92
  • Theorem 2.2
  • Theorem 5.1: PSW22
  • Proposition 7.1