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Equivariant Syzygies of the Ideal of 2 x 2 Permanents of a 2 x n Matrix

Jacob Zoromski

TL;DR

This work determines the equivariant syzygies of the ideal $P$ of $2\times 2$ subpermanents of a generic $2\times n$ matrix under natural group actions. By treating Ext^*_S(P, C) as a $G_n \times G_2$ representation and exploiting a pair of $G_n\times G_2$-equivariant short exact sequences, the authors show the minimal free resolution of $P$ has three linear strands and obtain explicit, strand-wise decompositions in terms of induced trivial, sign, and hook Specht modules. They provide closed-form formulas for the graded Betti numbers, including complete-intersection behavior when $n=3$ and a richer structure for $n>3$. The approach integrates exterior-algebra representation theory, Schur functors, and LR/Pieri-type induction to yield a comprehensive equivariant description that extends prior Betti-number results for permanents.

Abstract

We describe the equivariant syzygies of the ideal of $2 \times 2$ permanents of a generic $2 \times n$ matrix under its natural symmetric and torus group actions. Our proof gives us a new method of finding the Betti numbers of this ideal, which were first described by Gesmundo, Huang, Schenck, and Weyman.

Equivariant Syzygies of the Ideal of 2 x 2 Permanents of a 2 x n Matrix

TL;DR

This work determines the equivariant syzygies of the ideal of subpermanents of a generic matrix under natural group actions. By treating Ext^*_S(P, C) as a representation and exploiting a pair of -equivariant short exact sequences, the authors show the minimal free resolution of has three linear strands and obtain explicit, strand-wise decompositions in terms of induced trivial, sign, and hook Specht modules. They provide closed-form formulas for the graded Betti numbers, including complete-intersection behavior when and a richer structure for . The approach integrates exterior-algebra representation theory, Schur functors, and LR/Pieri-type induction to yield a comprehensive equivariant description that extends prior Betti-number results for permanents.

Abstract

We describe the equivariant syzygies of the ideal of permanents of a generic matrix under its natural symmetric and torus group actions. Our proof gives us a new method of finding the Betti numbers of this ideal, which were first described by Gesmundo, Huang, Schenck, and Weyman.

Paper Structure

This paper contains 8 sections, 12 theorems, 59 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Graded Betti Numbers
  3. Proof Outline
  4. Representation Structure on the Exterior Algebra
  5. Proof of Main Theorem
  6. Short Exact Sequences
  7. Equivariant Syzygies of $D$
  8. Equivariant Syzygies of $P$

Key Result

Theorem 1.1

The minimal free resolution of $P$ consists of three linear strands. The multidegrees $\mathbf{a}$ for which $\mathop{\mathrm{Ext}}\nolimits_S^\bullet(P,\mathbb{C})_{\mathbf{a}}$ is non-zero are permutations of $(2^a,1^b,0^{n-a-b})$ for various $a$ and $b$. The $\mathfrak{S}_n$-representations appea For all other $\mathbf{d}$, $\mathop{\mathrm{Ext}}\nolimits(P,\mathbb{C})_{\langle \mathbf{d} \rang

Theorems & Definitions (21)

  • Theorem 1.1
  • Example 1.2
  • Theorem 1.3: permanents, Thm. 1.2
  • Theorem 1.4
  • Example 2.1
  • Proposition 2.2
  • Proposition 3.1
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • ...and 11 more