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Interlacing networks: birational RSK, the octahedron recurrence, and Schur function identities. (English) Zbl 1315.05144
Summary: 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 Plücker-like three-term relations among path weights. We show that indeed these relations follow from the Plücker 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 M. Fulmek and M. Kleber [Electron. J. Comb. 8, No. 1, Research paper R16, 22 p. (2001; Zbl 0978.05005)]. These Schur function identities lead to some results on Schur positivity for expressions of the form \(s_\nu s_\rho - s_\lambda s_\mu\).

MSC:
05E10 Combinatorial aspects of representation theory
05E05 Symmetric functions and generalizations
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