Partial Algebraic Shifting

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 Erdős$\text{unicode}x2013$Ko$\text{unicode}x2013$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.