Product and puzzle formulae for GL_n Belkale-Kumar coefficients

TL;DR

This paper provides a combinatorial formula for Belkale-Kumar coefficients in type A flag manifolds, expressing them as products of Littlewood-Richardson numbers and introducing new puzzle-based methods.

Contribution

It introduces BK-puzzles that give a direct combinatorial formula for Belkale-Kumar structure constants, refining previous factorizations and relating to extremal honeycombs.

Findings

Belkale-Kumar coefficients are products of multiple Littlewood-Richardson numbers.

New puzzle constructions provide explicit combinatorial formulas.

Characterization of regular faces where Littlewood-Richardson numbers are always 1.

Abstract

The Belkale-Kumar product on H*(G/P) is a degeneration of the usual cup product on the cohomology ring of a generalized flag manifold. In the case G=GL_n, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that for G/P a (d-1)-step flag manifold, each Belkale-Kumar structure constant is a product of d(d-1)/2 Littlewood-Richardson numbers, for which there are many formulae available, e.g. the puzzles of [Knutson-Tao '03]. This refines previously known factorizations into d-1 factors. We define a new family of puzzles to assemble these to give a direct combinatorial formula for Belkale-Kumar structure constants. These "BK-puzzles" are related to extremal honeycombs, as in [Knutson-Tao-Woodward~'04]; using this relation we give another proof of Ressayre's result. Finally, we describe the regular faces of the Littlewood-Richardson cone on which the Littlewood-Richardson number is always 1; they correspond to nonzero Belkale-Kumar coefficients on partial flag manifolds where every subquotient has dimension 1 or 2.