Growth diagrams for the Schubert multiplication

We present a partial generalization to Schubert calculus on flag varieties of the classical Littlewood-Richardson rule, in its version based on Sch¨utzenberger’s jeu de taquin. More precisely, we describe certain structure constants expressing the product of a Schubert and a Schur polynomial. We use a generalization of Fomin’s growth diagrams (for chains in Young’s 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 S3-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin’s growth diagrams.

• Type: Journal
• Year: 2010
• Where: JCT
• Authors: Cristian Lenart
• Keywords: Fomin’s Growth, JCT 2010, K-Bruhat Order, Littlewood-Richardson Rule