On rotated Schur-positive sets

TL;DR

This paper proves that horizontal rotations of Schur-positive permutation sets remain Schur-positive, using cyclic actions on Young tableaux and jeu-de-taquin algorithms, advancing understanding of permutation set symmetries.

Contribution

The authors establish that horizontal rotations preserve Schur-positivity in permutation sets, introducing a cyclic descent set concept via Young tableaux and cyclic actions.

Findings

Horizontal rotations of Schur-positive sets are Schur-positive.

A cyclic action on Young tableaux is used to prove the main result.

A new cyclic descent set on tableaux is introduced and rotated.

Abstract

The problem of finding Schur-positive sets of permutations, originally posed by Gessel and Reutenauer, has seen some recent developments. Schur-positive sets of pattern-avoiding permutations have been found by Sagan et al and a general construction based on geometric operations on grid classes has been given by the authors. In this paper we prove that horizontal rotations of Schur-positive subsets of permutations are always Schur-positive. The proof applies a cyclic action on standard Young tableaux of certain skew shapes and a jeu-de-taquin type straightening algorithm. As a consequence of the proof we obtain a notion of cyclic descent set on these tableaux, which is rotated by the cyclic action on them.