Partial Algebraic Shifting

TL;DR

This paper investigates algebraic shifting of hypergraphs and simplicial complexes in exterior algebra with non-generic matrices, addressing open questions and identifying conditions for preserving Betti numbers, thus extending prior combinatorial shifting results.

Contribution

It introduces partial algebraic shifting in exterior algebra with non-generic matrices and provides conditions under which Betti numbers are preserved, linking to and generalizing combinatorial shifting.

Findings

Partial algebraic shifting extends previous combinatorial shifting.

A sufficient condition for Betti number preservation is identified.

Examples demonstrate the sharpness of the condition.

Abstract

We study algebraic shifting of uniform hypergraphs and finite simplicial complexes in the exterior algebra with respect to matrices which are not necessarily generic. Several questions raised by Kalai (2002) are addressed. For instance, it turns out that the combinatorial shifting of Erd\H{o}s$\unicode{x2013}$Ko$\unicode{x2013}$Rado (1961) arises as a special case. Moreover, we identify a sufficient condition for partial shifting to preserve the Betti numbers of a simplicial complex; examples show that this condition is sharp.