Resolutions of letterplace ideals of posets

TL;DR

This paper studies the resolutions of letterplace ideals of posets, providing topological methods to compute their Betti numbers and offering recursive procedures for specific poset structures.

Contribution

It introduces new topological techniques for calculating Betti numbers of letterplace ideals and offers recursive methods for posets with tree-structured Hasse diagrams.

Findings

Betti numbers can be computed from simplicial complexes with no more than c vertices for posets union of c chains.

Recursive procedure developed for Betti diagram computation when Hasse diagram has a tree structure.

Structural results on multigraded Betti numbers of letterplace ideals.

Abstract

We investigate resolutions of letterplace ideals of posets. We develop topological results to compute their multigraded Betti numbers, and to give structural results on these Betti numbers. If the poset is a union of no more than $c$ chains, we show that the Betti numbers may be computed from simplicial complexes of no more than $c$ vertices. We also give a recursive procedure to compute the Betti diagrams when the Hasse diagram of $P$ has tree structure.