2 x 2 permanental ideals of hypermatrices

TL;DR

This paper investigates the algebraic structure of 2x2 permanental ideals in hypermatrices, extending previous work on generic matrices and comparing permanental and determinantal ideals, revealing increased complexity in permanental cases.

Contribution

It introduces a detailed analysis of 2x2 permanental ideals of hypermatrices, generalizing prior results and highlighting differences from determinantal ideals, including the role of t-switchability and minimal primes.

Findings

Permanental ideals require additional restrictions compared to determinantal ideals.

The structure of permanental ideals is more complex, with more minimal primes.

Comparison with determinantal ideals reveals key differences in algebraic properties.

Abstract

We study the structure of ideals generated by some classes of 2 \times 2 permanents of hypermatrices. This generalizes [9] on 2 x 2 permanental ideal of generic matrices. We compare the obtained structure to that of the corresponding determinantal ideals in [Swanson-Taylor 11]: while the notion of t-switchability introduced in [11] plays a role for both permanental and determinantal ideals, the permanents require further restrictions, which in general increases the number of minimal primes. In the last two section we examine a few related classes of permanental ideals. This is an extension of the first author's senior thesis at Reed College, 2011, under the second author's supervision.