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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:
 5e+10 Combinatorial aspects of representation theory 500000 Symmetric functions and generalizations
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##### References:
 [1] Bergeron, François; Biagioli, Riccardo; Rosas, Mercedes H., Inequalities between Littlewood-Richardson coefficients, J. Combin. Theory Ser. A, 113, 4, 567-590, (2006) · Zbl 1090.05070 [2] Chari, Vyjayanthi; Fourier, Ghislain; Sagaki, Daisuke, Posets, tensor products and Schur positivity, Algebra Number Theory, 8, 4, 933-961, (2014) · Zbl 1320.17004 [3] Corwin, Ivan; O’Connell, Neil; Seppäläinen, Timo; Zygouras, Nikolaos, Tropical combinatorics and Whittaker functions, Duke Math. J., 163, 3, 513-563, (2014) · Zbl 1288.82022 [4] Danilov, Vladimir I.; Karzanov, Alexander V.; Koshevoy, Gleb A., Planar flows and quadratic relations over semirings, J. Algebraic Combin., 36, 3, 441-474, (2012) · Zbl 1254.05071 [5] Danilov, V. I.; Koshevoy, G. A., Arrays and combinatorics of Young tableaux, Uspekhi Mat. Nauk, Russian Math. Surveys, 60, 2, 269-334, (2005), English translation in · Zbl 1081.05104 [6] Danilov, V. I.; Koshevoy, G. A., Arrays and the octahedron recurrence, (April 2005) [7] Danilov, V. I.; Koshevoy, G. A., The octahedron recurrence and RSK-correspondence, Sém. Lothar. Combin., 54A, (2005/07), Art. B54An, 16 pp. (electronic) · Zbl 1267.05292 [8] Dobrovolska, Galyna; Pylyavskyy, Pavlo, On products of $$\mathfrak{sl}_n$$ characters and support containment, J. Algebra, 316, 2, 706-714, (2007) · Zbl 1130.17004 [9] Fomin, Sergey; Fulton, William; Li, Chi-Kwong; Poon, Yiu-Tung, Eigenvalues, singular values, and Littlewood-Richardson coefficients, Amer. J. Math., 127, 1, 101-127, (2005) · Zbl 1072.15010 [10] Freese, Ralph, An application of Dilworth’s lattice of maximal antichains, Discrete Math., 7, 107-109, (1974) · Zbl 0271.05011 [11] Fulmek, Markus; Kleber, Michael, Bijective proofs for Schur function identities which imply Dodgson’s condensation formula and Plücker relations, Electron. J. Combin., 8, 1, (2001), Research Paper 16, 22 pp. (electronic) · Zbl 0978.05005 [12] Greene, Curtis, An extension of Schensted’s theorem, Adv. Math., 14, 254-265, (1974) · Zbl 0303.05006 [13] Gurevich, Dimitri; Pyatov, Pavel; Saponov, Pavel, Bilinear identities on Schur symmetric functions, J. Nonlinear Math. Phys., 17, suppl. 1, 31-48, (2010) · Zbl 1362.05129 [14] Henriques, André, A periodicity theorem for the octahedron recurrence, J. Algebraic Combin., 26, 1, 1-26, (2007) · Zbl 1125.05106 [15] Hopkins, S., RSK via local transformations, (April 2014) [16] Kirillov, A. N., Completeness of states of the generalized Heisenberg magnet, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 134, 169-189, (1984), Automorphic functions and number theory, II · Zbl 0538.58016 [17] Kirillov, Anatol N., Introduction to tropical combinatorics, (Physics and Combinatorics, Nagoya, 2000, (2001), World Sci. Publ. River Edge, NJ), 82-150 · Zbl 0989.05127 [18] Krattenthaler, C., Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes, Adv. in Appl. Math., 37, 3, 404-431, (2006) · Zbl 1108.05095 [19] Lam, Thomas; Postnikov, Alexander; Pylyavskyy, Pavlo, Schur positivity and Schur log-concavity, Amer. J. Math., 129, 6, 1611-1622, (2007) · Zbl 1131.05096 [20] Lindström, Bernt, On the vector representations of induced matroids, Bull. Lond. Math. Soc., 5, 85-90, (1973) · Zbl 0262.05018 [21] Lusztig, George, Introduction to total positivity, (Positivity in Lie Theory: Open Problems, de Gruyter Exp. Math., vol. 26, (1998), de Gruyter Berlin), 133-145 · Zbl 0929.20035 [22] McNamara, Peter R. W., Necessary conditions for Schur-positivity, J. Algebraic Combin., 28, 4, 495-507, (2008) · Zbl 1160.05057 [23] Noumi, Masatoshi; Yamada, Yasuhiko, Tropical Robinson-Schensted-Knuth correspondence and birational Weyl group actions, (Representation Theory of Algebraic Groups and Quantum Groups, Adv. Stud. Pure Math., vol. 40, (2004), Math. Soc. Japan Tokyo), 371-442 · Zbl 1061.05103 [24] O’Connell, Neil, Geometric RSK and the Toda lattice, Illinois J. Math., 57, 3, 883-918, (2013) · Zbl 1325.37047 [25] O’Connell, Neil; Seppäläinen, Timo; Zygouras, Nikos, Geometric RSK correspondence, Whittaker functions and symmetrized random polymers, Invent. Math., 197, 2, 361-416, (2014) · Zbl 1298.05323 [26] Postnikov, Alexander, Total positivity, Grassmannians, and networks, (September 2006) [27] Schilling, Anne; Warnaar, S. Ole, Inhomogeneous lattice paths, generalized kostka polynomials and $$A_{n - 1}$$ supernomials, Comm. Math. Phys., 202, 2, 359-401, (1999) · Zbl 0935.05090 [28] Shimozono, Mark, Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebraic Combin., 15, 2, 151-187, (2002) · Zbl 1106.17305 [29] Shimozono, Mark; White, Dennis E., A color-to-spin domino Schensted algorithm, Electron. J. Combin., 8, 1, (2001), Research Paper 21, 50 pp. (electronic) · Zbl 0965.05095 [30] Skandera, Mark, Inequalities in products of minors of totally nonnegative matrices, J. Algebraic Combin., 20, 2, 195-211, (2004) · Zbl 1066.05089 [31] Speyer, David E., Perfect matchings and the octahedron recurrence, J. Algebraic Combin., 25, 3, 309-348, (2007) · Zbl 1119.05092 [32] Stanley, Richard P., Enumerative combinatorics, vol. 2, (1999), Cambridge University Press Cambridge · Zbl 0928.05001 [33] Stanley, Richard P., Enumerative combinatorics, vol. 1, (2012), Cambridge University Press Cambridge · Zbl 1247.05003 [34] Stembridge, John R., Multiplicity-free products of Schur functions, Ann. Comb., 5, 2, 113-121, (2001) · Zbl 0990.05130 [35] X. Viennot, I.M. Gessel, Determinants, paths, and plane partitions, preprint, 1988.
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