Product and Puzzle Formulae for GLn Belkale-Kumar Coefficients

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 = GLn, it was used by N. Ressayre to determine the regular faces of the Littlewood-Richardson cone. We show that forG/P a (d−1)-step flagmanifold, each Belkale-Kumar structure constant is a product of ( d 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.