Reverse lexicographic and lexicographic shifting

Abstract

A short new proof of the fact that all shifted complexes are fixed by reverse lexicographic shifting is given. A notion of lexicographic shifting, Δ lex - an operation that transforms a monomial ideal of S = k[x i,: i ∈ ℕ] that is finitely generated in each degree into a squarefree strongly stable ideal - is defined and studied. It is proved that (in contrast to the reverse lexicographic case) a squarefree strongly stable ideal I ⊂ S is fixed by lexicographic shifting if and only if I is a universal squarefree lexsegment ideal (abbreviated USLI) of S. Moreover, in the case when I is finitely generated and is not a USLI, it is verified that all the ideals in the sequence {Δ lexi(I)} i=0∞ are distinct. The limit ideal Δ̄(I) = lim i → ∞ Δ(I) is well defined and is a USLI that depends only on a certain analog of the Hubert function of I. © Springer Science + Business Media, Inc. 2006.