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, Δlex—an operation that transforms a monomial ideal of S = K[xi: 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 $$\{ \Delta_{\rm lex}^{i} (I) \}_{i=0}^{\infty}$$ } are distinct. The limit ideal $$\bar{\Delta}(I) = {\rm lim}_{i \rightarrow \infty} \Delta_{\rm lex}^{i} (I)$$ is well defined and is a USLI that depends only on a certain analog of the Hilbert function of I.