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The main reasons for matrices multiplying to zero

Jakub Koncki, Richard Rimanyi

TL;DR

This work analyzes the zero-product variety $\Sigma_{\underline{d}}$ for a dimension vector $\underline{d}$, focusing on its maximal-dimensional components. It encodes these components via a quadratic integer program (QIP) whose optimal solutions index the components, and introduces lace diagrams that simultaneously visualize generic elements of each component and their defining equations. Building on LR’s explicit formulas for the codimension $C$ and the count $\theta$ of top components, the authors prove a bijection between optimal QIP solutions and maximal-dimensional orbit closures, using Ext-dimension calculations through the Voight lemma. The resulting framework provides a concrete, diagrammatic classification of the main reasons for matrices multiplying to zero, with potential implications for applications in learning theory and quiver representations.

Abstract

We provide an explicit description of the maximal-dimensional components of the variety parametrizing sequences of matrices of prescribed sizes whose product is zero.

The main reasons for matrices multiplying to zero

TL;DR

This work analyzes the zero-product variety for a dimension vector , focusing on its maximal-dimensional components. It encodes these components via a quadratic integer program (QIP) whose optimal solutions index the components, and introduces lace diagrams that simultaneously visualize generic elements of each component and their defining equations. Building on LR’s explicit formulas for the codimension and the count of top components, the authors prove a bijection between optimal QIP solutions and maximal-dimensional orbit closures, using Ext-dimension calculations through the Voight lemma. The resulting framework provides a concrete, diagrammatic classification of the main reasons for matrices multiplying to zero, with potential implications for applications in learning theory and quiver representations.

Abstract

We provide an explicit description of the maximal-dimensional components of the variety parametrizing sequences of matrices of prescribed sizes whose product is zero.

Paper Structure

This paper contains 14 sections, 13 theorems, 45 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Equioriented type A quiver representations and their orbits
  3. The quiver representation
  4. Orbits
  5. Quiver modules
  6. Voight lemma
  7. $\Sigma_{\underline{d}}$ and its components
  8. The protagonist
  9. Known results on $C$, $\theta$
  10. Description of the maximal-dimensional components, in a special case
  11. Description of the maximal-dimensional components of $\Sigma_{\underline{d}}$
  12. Proof
  13. The case $k=0$
  14. The general case

Key Result

Lemma 2.2

voight A normal slice to $\mathcal{O}_{\underline{m}}$ at a point is isomorphic to $\mathop{\mathrm{Ext}}\nolimits(\mathop{\mathrm{M}}\nolimits_{\underline{m}},\mathop{\mathrm{M}}\nolimits_{\underline{m}})$.

Figures (4)

  • Figure 1: A Kostant partition $\underline{m}$ for the dimension vector $\underline{d}=(8,7,5,9,5,8)$, the induced rank pattern $\underline{r}$ (row and column indices run from 0 to $n$), and a lace diagram representing $\underline{m}$. Another lace diagram representing the same $\underline{m}$ is the top-left picture in Figure \ref{['fig:875958 raising vectors']}.
  • Figure 2: The optimal lace diagrams for dimension vector $\underline{d}=(5,5,7,8,8,9)$ in Theorem \ref{['thm:components']}. They correspond to the $\theta=4$ solutions of \ref{['eq:QIP']}: $(4,1,0,0,0)$ (top left), $(3,2,0,0,0)$ (top right), $(3,1,1,0,0)$ (bottom left), $(3,1,0,1,0)$ (bottom right).
  • Figure 3: The lace diagrams associated to the dimension vector $\underline{d}=(5,5,7,8,8,9)$, and raising vectors $(\star,4,1,0,0,0)$ (top left), $(\star,3,2,0,0,0)$ (top right), $(\star,3,1,1,0,0)$ (bottom left), $(\star,3,1,0,1,0)$ (bottom right). Notice that these lace diagrams encode the same Kostant partitions as the ones in Figure \ref{['fig:557889_original']}. Also, these raising vectors are the solutions of (QIP'), hence these lace diagrams describe the maximal-dimensional components of $\Sigma_{\underline{d}}$.
  • Figure 4: The lace diagrams associated to the dimension vector $\underline{d}=(8,7,5,9,5,8)$, and raising vectors $(0,1,\star,0,4,0)$ (top left), $(0,2,\star,0,3,0)$ (top right), $(0,1,\star,0,3,1)$ (bottom left), $(1,1,\star,0,3,0)$ (bottom right). In fact, these raising vectors are the solutions of (QIP'), hence these lace diagrams describe the maximal-dimensional components of $\Sigma_{\underline{d}}$.

Theorems & Definitions (33)

  • Definition 2.1
  • Lemma 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Corollary 2.5
  • Definition 3.1
  • Definition 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • ...and 23 more