Schur-positive sets of permutations via products and grid classes (Q517354)

Given a subset \(A\) of the symmetric group \(S_n\), the associated quasisymmetric function is \({\mathcal Q}(A)= \sum_{\pi\in A}F_{n,\text{Des}(\pi)}\), where \(\text{Des}(\pi)\) is the set of descents, and \(F_{n,\text{Des}(\pi)}\) is \textit{I. M. Gessel}'s fundamental quasisymmetric function [Contemp.\ Math.\ 34. 289--302 (1984, Zbl.\ 0562.05007)]. It is a long-standing question to find classes of sets \(A\) for which \({\mathcal Q}(A)\) is Schur-positive. The authors construct Schur-positive sets, in which permutations are characterized by geometric grid classes. A geometric grid corresponds to a matrix with entries \(-1\), \(0\), \(1\), in which we draw lines of slope \(\pm 1\) for each \(1\) and \(-1\) in the matrix, and a blank space for each \(0\). A permutation is in a geometric grid class if its entries can be written as dots on these lines, preserving ordering. For example, the grid class \(\binom{-1}{1}\), corresponding to the shape \(<\), contains the left-unimodal permutations, in which every prefix is an interval in \(\mathbb Z\). Results on these grid classes give many new classes of Schur-positive sets and multisets, and combine many individual results into a more general framework.