Interlacing networks: birational RSK, the octahedron recurrence, and Schur function identities

TL;DR

This paper introduces interlacing networks to provide a bijective proof of the octahedron recurrence in birational RSK, connecting path relations with Grassmannian and Schur function identities.

Contribution

It defines interlacing networks and shows their relation to the octahedron recurrence, Schur identities, and the totally nonnegative Grassmannian, offering new combinatorial and algebraic insights.

Findings

Interlacing networks satisfy Plücker-like relations.

Octahedron recurrence follows from network path relations.

Schur function identities imply Schur positivity results.

Abstract

Motivated by the problem of giving a bijective proof of the fact that the birational RSK correspondence satisfies the octahedron recurrence, we define interlacing networks, which are certain planar directed networks with a rigid structure of sources and sinks. We describe an involution that swaps paths in these networks and leads to Pl\"{u}cker-like three-term relations among path weights. We show that indeed these relations follow from the Pl\"{u}cker relations in the Grassmannian together with some simple rank properties of the matrices corresponding to our interlacing networks. The space of matrices obeying these rank properties forms the closure of a cell in the matroid stratification of the totally nonnegative Grassmannian. Not only does the octahedron recurrence for RSK follow immediately from the three-term relations for interlacing networks, but also these relations imply some interesting identities of Schur functions reminiscent of those obtained by Fulmek and Kleber. These Schur function identities lead to some results on Schur positivity for expressions of the form $s_{\nu}s_{\rho} - s_{\lambda}s_{\mu}$.