Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals

TL;DR

This paper investigates chains of Laurent lattice ideals invariant under symmetric group actions, proving local stabilization up to symmetry and providing an algorithm for explicit stabilization generators, with applications in algebraic statistics.

Contribution

It introduces a new stabilization theorem for symmetric invariant chains of Laurent lattice ideals and provides an implemented algorithm for constructing stabilization generators.

Findings

Chains of symmetric Laurent lattice ideals stabilize locally up to symmetry.

An explicit algorithm for constructing stabilization generators is developed and implemented.

Applications demonstrated in algebraic statistics.

Abstract

We study chains of lattice ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a theorem which says that "up to the action of the group", these chains locally stabilize. We also give an algorithm, which we have implemented in software, for explicitly constructing these stabilization generators for a family of Laurent toric ideals involved in applications to algebraic statistics. We close with several open problems and conjectures arising from our theoretical and computational investigations.