Equivariant lattice bases

TL;DR

This paper investigates infinite-rank lattices invariant under symmetric group actions, introducing a new approach to establish finiteness of their equivariant bases, with applications in algebraic statistics.

Contribution

It presents a novel method for proving finiteness of equivariant bases in invariant lattices, extending previous results in algebraic statistics.

Findings

Every invariant lattice in ^{(\u220fa7a7[c])} has a finite equivariant Graver basis

Generalizes previous finiteness results on Markov bases

Provides a new framework for algebraic statistical methods

Abstract

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in $\mathbb{Z}^{(\mathbb{N}\times[c])}$, where $c\in\mathbb{N}$, has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature.