The octahedron recurrence and RSK-correspondence

TL;DR

This paper explores the algebraic and tropical versions of the octahedron recurrence related to the RSK correspondence, establishing several bijections between arrays, functions, and plane partitions.

Contribution

It introduces a novel algebraic and tropical framework for the RSK correspondence using octahedron recurrence and establishes new bijections among arrays, functions, and plane partitions.

Findings

Established a bijection between non-negative arrays and supermodular functions.

Derived a piecewise linear bijection between supermodular and infra-modular functions.

Connected infra-modular functions to plane partitions via linear bijections.

Abstract

We start with an ``algebraic'' RSK-correspondence due to Noumi and Yamada. Given a matrix $X$, we consider a pyramidal array of solid minors of $X$. It turns out that this array satisfies an algebraic variant of octahedron recurrence. The main observation is that this array can also be constructed with the help of some square `genetic' array. Next we tropicalize this algebraic construction and consider $T$-{\em polarized} pyramidal arrays (that is arrays satisfying octahedral relations). As a result we get several bijections, viz: a) a linear bijection between non-negative arrays and supermodular functions; b) a piecewise linear bijection between supermodular functions and the so called infra-modular functions; c) a linear bijection between infra-modular functions and plane partitions. A composition of these bijections yields a bijection between non-negative arrays and plane partitions coinciding with the modified RSK-correspondence.