Schur-positive sets of permutations via products of grid classes

TL;DR

This paper introduces a geometric grid class-based method for constructing Schur-positive permutation sets, broadening understanding and generating new examples in algebraic combinatorics.

Contribution

It provides a general framework using grid classes and product operations to systematically create Schur-positive sets, explaining previously sporadic cases.

Findings

Many new Schur-positive sets are constructed.

The framework explains the existence of known sporadic Schur-positive sets.

The method unifies and extends previous constructions.

Abstract

Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur-positive is a long-standing problem in algebraic combinatorics. In this paper we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets, and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.