Properties of the nonsymmetric Robinson-Schensted-Knuth algorithm

We introduce a generalization of the Robinson-Schensted-Knuth algorithm to composition tableaux involving an arbitrary permutation. If the permutation is the identity our construction reduces to Mason’s original composition Robinson-Schensted-Knuth algorithm. In particular we develop an analogue of Schensted insertion in our more general setting, and use this to obtain new decompositions of the Schur function into nonsymmetric elements (which become Demazure atoms when the permutation is the identity). Other applications include Pieri rules for multiplying these generalized Demazure atoms by complete homogeneous symmetric functions or elementary symmetric functions, a generalization of Knuth’s correspondence between matrices of nonnegative integers and pairs of tableaux, and a version of evacuation for arbitrary permutations.