Permanental ideals of symmetric matrices

TL;DR

This paper investigates the algebraic properties of the ideal generated by 2x2 permanents of symmetric matrices, including its Gr"obner basis, dimension, and primary decomposition, highlighting dependence on the base field's characteristic.

Contribution

It provides a comprehensive analysis of the permanental ideal of symmetric matrices, including explicit computations and primary decomposition, emphasizing characteristic dependence.

Findings

Computed Gr"obner basis of the ideal

Determined dimension and depth of the ideal

Established primary decomposition and dependence on characteristic

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 Gr\"obner 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.