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

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:133

Article

Farber, Miriam¹  Hopkins, Sam¹  Trongsiriwat, Wuttisak¹ 

[1] MIT, Cambridge, MA 02139 USA

关键词: Lindstrom-Gessel-Viennot;    Plucker relations;    RSK;    Octahedron recurrence;    Schur functions;    Schur positivity;   

DOI  :  10.1016/j.jcta.2015.03.001

来源: Elsevier

【 摘 要 】

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 Plucker-like three-term relations among path weights. We show that indeed these relations follow from the Plucker 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). (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free