Introduction to tropical combinatorics (Q2751090)

This paper constructs birational representations of the symmetric, affine symmetric and extended affine symmetric groups on the affine spaces \(A^{n^2}\) and \(A^{n(n+1)/2}\). It is shown that the tropical version of RSK (Robinson-Schensted-Knuth correspondence) has many properties which are similar to those of the classical RSK. Tropical version of statistics cocharge \(\overline c_n\) has been studied and it is proved that it is invariant with respect to the birational action of the symmetric group on the space \(A^{n(n+1)/2}\) constructed in this paper. It is also proved that the tropical version of Schützenberger's involution satisfies the discrete Hirota-Miwa type equations (the Plücker relations for the \(q\)th variation of Schur functions). This result allows to find an explicit formula for the tropical version of Schützenberger's involution as a generating function for certain weighted non-crossing lattice paths.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00051].