A simple definition for the universal Grassmannian order
This paper is devoted to a simple combinatorial description of a partial order, the Grassmannian-Bruhat order (\(<\)) on the symmetric group \(S_n\) introduced by \textit{N. Bergeron} and the reviewer to study the problem of multiplying Schubert polynomials [Duke Math. J. 95, 373-423 (1998; Zbl 0939.05084)]. Its main results include a new and simpler definition for this order which characterizes it as as the meet in the poset of partial orders on \(S_n\) of the usual weak order and one given by a certain permutation statistic, which turns out to be the rank function for \(<\). Most properties of \(<\) (including those important for loc. cit.) follow quite directly from this characterization, simplifying the original arguments. The final section studies the rank-generating function for \(<\), showing that it is divisible by \((1+t)^{\lfloor\frac{n}{2}\rfloor}\). Two early typos (both omitting `\(>\)') may impede the understanding of the reader: The end of line 8 of Section 1 (p.~348) should be `and \(\sigma[i]>\sigma[j]\), and'. Also, the right hand side of first displayed formula on page 349 should be `\(\{i\mid \sigma[i]>i\}\)'.
scientific article; zbMATH DE number 1945002
scientific article