Promotion permutations for tableaux

TL;DR

This paper introduces the concept of promotion permutations and matrices within the broader context of fluctuating tableaux, linking combinatorics with algebraic operations and extending previous tableau classes.

Contribution

It develops new notions of promotion permutations and matrices, and generalizes the theory to fluctuating tableaux, unifying various classes of tableaux and supporting algebraic constructions.

Findings

Introduction of promotion permutations and matrices for tableaux

Development of combinatorics and representation theory of fluctuating tableaux

Connection between web rotation and promotion action on tableaux

Abstract

In our companion paper, we develop a new $SL_4$-web basis. Basis elements are given by certain planar graphs and are constructed so that important algebraic operations can be performed diagrammatically. A guiding principle behind our construction is that the long cycle $(12\ldots n) \in \mathfrak{S}_n$ should act by rotation of webs. Moreover, the bijection between webs and tableaux should intertwine rotation with the promotion action on tableaux. In this paper, we develop necessary notions of promotion permutations and promotion matrices, which are new even for standard tableaux. To support inductive arguments in the companion paper, we must however work in the more general setting of fluctuating tableaux, which we introduce and which subsumes many classes of tableaux that have been previously studied, including (generalized) oscillating, vacillating, rational, alternating, and (semi)standard tableaux. Therefore, we also give here a full development of the basic combinatorics and representation theory of fluctuating tableaux.