Prime splittings of Determinantal Ideals

TL;DR

This paper investigates determinantal ideals encoded by hypergraphs, identifying conditions for Gr"obner bases, and describing algebraic invariants and prime decompositions through combinatorial hypergraph data.

Contribution

It provides a combinatorial framework for understanding prime splittings and resolutions of determinantal ideals associated with hypergraphs.

Findings

Conditions for Gr"obner bases of determinantal ideals

Description of algebraic invariants via hypergraph clique decompositions

Prime splitting characterizations for specific hypergraph classes

Abstract

We consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gr\"obner basis. In this case, the ideal is radical, and we can describe algebraic and numerical invariants of these ideals in terms of combinatorial data of their hypergraphs, such as the clique decomposition. In particular, we can construct a minimal free resolution as a tensor product of the minimal free resolution of their cliques. For several classes of hypergraphs we find a combinatorial description of the minimal primes in terms of a prime splitting. That is, we write the determinantal ideal as a sum of smaller determinantal ideals such that each minimal prime is a sum of minimal primes of the summands.