Growth Diagrams for the Schubert Multiplication

TL;DR

This paper extends classical Littlewood-Richardson rules to Schubert calculus on flag varieties using growth diagrams, enabling new combinatorial computations of structure constants involving Schubert and Schur polynomials.

Contribution

It introduces a partial generalization of growth diagrams to chains in k-Bruhat order, connecting Schubert calculus with combinatorial structures.

Findings

Describes structure constants for Schubert and Schur polynomial products.

Generalizes Fomin's growth diagrams to k-Bruhat chains.

Potentially extends S_3-symmetric Littlewood-Richardson rule.

Abstract

We present a partial generalization to Schubert calculus on flag varieties of the classical Littlewood-Richardson rule, in its version based on Schuetzenberger'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 S_3-symmetric Littlewood-Richardson rule due to Thomas and Yong, which is based on Fomin's growth diagrams.