A combinatorial construction of the Schubert polynomials
Let \(w=(w_ 1,w_ 2,\dots,w_ n)\) be a permutation in the symmetric group \(S_ n\). An explicit combinatorial construction of the Schubert polynomial \({\mathcal S}_ w\) is given as a sum of weights \(x^ D\) of diagrams \(D\) (finite nonempty sets of lattice points \((i,j)\) in the positive quadrant), where the sum is over the set \(\Omega(w)\) of diagrams \(D\) which are obtainable from an initial diagram \(D(w)\) by repeated application of allowable \(B\)-moves. If we view a diagram as a set of checkers (or black stones for those who prefer) distributed over the positive quadrant, then a \(B\)-move is just a move of one checker from a position to an adjacent unoccupied position subject to a set of rules, thereby obtaining a new diagram. Since the classical Schur functions can be viewed as special cases of the Schubert polynomials for appropriate choices of the permutation \(w\), it has been hoped that there was a combinatorial algorithm for constructing Schubert polynomials which extends (or is at least in analogy to) the construction of the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\) as a sum of weights of column strict tableaux of shape \(\lambda\), where \(\lambda\) is a partition. The algorithm given here is not quite an extension of the usual construction for Schur functions, however a simple bijection between \(\Omega(w)\) and the set of column strict tableaux of shape \(\lambda\) is given in the particular case where the Schubert polynomial \({\mathcal S}_ w\) is the Schur function \(S_ \lambda(x_ 1,\dots,x_ r)\). A different algorithm for constructing the general Schubert polynomial \({\mathcal S}_ w\) was conjectured (and proved when \(w\) is a vexillary permutation) by \textit{A. Kohnert} [Weintrauben, Polynome, Tableaux, Bayreuther Math. Schr. 38, 1-97 (1991; Zbl 0755.05095)]. A sketch is given at the end of the paper here of how one would show the equivalence of the rule given here and that of Kohnert.
scientific article; zbMATH DE number 64857
scientific article