Abstract: Physics and Combinatorics, pp. 82-150 (2001) No AccessINTRODUCTION TO TROPICAL COMBINATORICSANATOL N. KIRILLOVANATOL N. KIRILLOVGraduate School of Mathematics, Nagoya University, Chikusa–ku, Nagoya 486–8602, JapanSteklov Mathematical Institute at St.Petersburg, Fontanka 27, St.Petersburg 191011, Russiahttps://doi.org/10.1142/9789812810007_0005Cited by:28 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: We construct birational representations of the symmetric, affine symmetric and extended affine symmetric groups on the affine spaces 𝔸n2 and 𝔸n(n+1)/2 which are tropical versions of the action of these groups on the set of transportation matrices and Lascoux–Schützenberger's action of the symmetric group on the set of semistandard Young tableaux. We show that tropical version of the Robinson–Schensted–Knuth correspondence (RSK for short) is a birational automorphism of the affine space 𝔸n2 which intertwines two birational actions of the symmetric group on the space 𝔸n2. We show that tropical version of RSK has many properties which are similar to those of the classical RSK. We study tropical version of statistics cocharge and proof, in particular, that it is invariant with respect to the birational action of the symmetric group on the space 𝔸n(n + 1)/2 constructed in this paper. We prove that tropical version of Schützenberger's involution satisfies the discrete Hirota–Miwa type equations (the Plücker relations for the 9-th variation of Schur functions). This result allows to find explicit formula for the tropical version of Schützenberger's involution as a generating function for certain weighted non–crossing lattice paths. 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Publication Year: 2001
Publication Date: 2001-04-01
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 99
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