Arrays and the octahedron recurrence

TL;DR

This paper explores the combinatorics of arrays related to the octahedron recurrence, establishing connections with Young tableaux, discrete convexity, and the RSK correspondence, and confirms conjectures linking these structures.

Contribution

It unifies various bijections involving arrays, hives, and Young tableaux through the octahedron recurrence, confirming conjectures and revealing new combinatorial insights.

Findings

Bijections of associativity and commutativity arise naturally in array theory.

The octahedron recurrence coincides with discrete convexity.

A new bijection using the octahedron recurrence and RSK correspondence is constructed.

Abstract

Recently, in papers by Knutson, Tao and Woodward, Henriques and Kamnitzer, Pak and Vallejo have been constructed several interesting bijections of associativity and commutativity. In the first two papers bijections relate special sets of discretely concave functions (hives) on triangular grids and the octahedron recurrence plays the key role for these bijections. Pak and Vallejo related special sets of Young tableaux and constructions of these bijections based on standard algorithms in this theory, jeu de taquen, Schutzenberger involution, tableaux switching, etc. In this paper we investigate these constructions from the third point of view, combinatorics of arrays, theory worked out recently by the authors. Arrays naturally related as well to functions on the lattice of integers as to Young tableaux. In the tensor category of arrays, the bijections of associativity and commutativity arise naturally. We establish coincidence of our bijections with that defined in the first two papers and in the integer-valued set-up with the bijection in the third paper (that is, in particular, a solution of Conjecture 1 by Pak and Vallejo). In order to relate different approaches and to reveal combinatorics of the octahedron recurrence, we, first, show that the octahedron recurrence agrees with discrete convexity and, second, we construct another bijection using the octahedron recurrence, the functional form of the RSK correspondence.