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On the algebra generated by three commuting matrices: combinatorial cases

Ron Cherny, Matthew Satriano, Yohan Song

Abstract

Gerstenhaber proved in 1961 that the unital algebra generated by a pair of commuting $d\times d$ matrices over a field has dimension at most $d$. It is an open problem whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for special classes of triples of matrices arising from combinatorial data.

On the algebra generated by three commuting matrices: combinatorial cases

Abstract

Gerstenhaber proved in 1961 that the unital algebra generated by a pair of commuting matrices over a field has dimension at most . It is an open problem whether the analogous statement is true for triples of matrices which pairwise commute. We answer this question for special classes of triples of matrices arising from combinatorial data.
Paper Structure (6 sections, 11 theorems, 34 equations)

This paper contains 6 sections, 11 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Combinatorial Gerstenhaber problem and Young diagrams
  3. Restricting the shape of counter-examples: scaffolded towers
  4. Reducing to two-dimensional data: floor plans
  5. Constraints on the border of a floor plan
  6. Proof of Theorem \ref{['thm:main-intro']}

Key Result

Theorem 1.1

If $S=k[x_1,x_2,x_3]$ and $N=\bigoplus_i S/(x_1,x_2,x_3^{n_i})$, then the combinatorial Gerstenhaber problem holds when gluing along $N$.

Theorems & Definitions (41)

  • Example 1.1
  • Definition 1.1
  • Remark 1.1
  • Definition 1.2
  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • proof
  • Example 2.1
  • ...and 31 more