Permanental ideals of symmetric matrices

TL;DR

The paper investigates the permanental ideal $P_2(X)$ generated by the $2\times 2$ permanents of a symmetric $n\times n$ matrix $X$ over a field with $\mathrm{char}(\Bbbk)\neq 2$. It develops a diagonal Gröbner basis, determines all minimal primes $P_{ij}$, and provides an explicit, irredundant primary decomposition that reveals multiple embedded primes beyond the maximal ideal. The results show that, unlike the determinantal case, $P_2(X)$ exhibits characteristic-dependent algebraic properties, including non-Groebner-generators for certain presentations and a non-Cohen–Macaulay coordinate ring for $n>2$. These findings illuminate the distinct structural behavior of permanental ideals in the symmetric setting and furnish precise tools for their algebraic and geometric analysis.

Abstract

In this article, we study the ideal generated by $2\times 2$ permanents of a symmetric matrix. We denote this ideal by $P_2(X)$ where $X$ is a symmetric matrix. We compute a Gröbner basis, dimension, depth, minimal primes, and a primary decomposition of $P_2(X)$. It can be seen that the answer is reliant on whether the characteristic of the base field is two, and thus these ideals constitute a class of ideals whose algebraic properties depend on characteristics of the base field.

Theorems & Definitions (31)

• Lemma 2.1: LS00
• proof
• Lemma 2.2: LS00
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 3.1
• proof : Proof of Theorem \ref{['thm:Grobner-symmetric']}