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{\bf Allen Knutson and Kevin Purbhoo}
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{\bf Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients}
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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 $d\choose 2$
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 $2$.

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