Growth diagrams for the Schubert multiplication

We present a partial generalization of the classical Littlewood–Richardson rule (in its version based on Schützenbergerʹs jeu de taquin) to Schubert calculus on flag varieties. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fominʹs growth diagrams (for chains in Youngʹs lattice of partitions) to chains of permutations in the so-called k-Bruhat order. Our work is based on the recent thesis of Beligan, in which he generalizes the classical plactic structure on words to chains in certain intervals in k-Bruhat order. Potential applications of our work include the generalization of the S 3 -symmetric Littlewood–Richardson rule due to Thomas and Yong, which is based on Fominʹs growth diagrams.