Geodesic
Product and puzzle formulae for \(GL_n\) Belkale-Kumar coefficients
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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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 $2$.
DOI : 10.37236/563
Classification : 05A15, 05B45, 14M15
Mots-clés : Belkale-Kumar product, combinatorial formula for Belkale-Kumar structure constants 
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     author = {Allen Knutson and Kevin Purbhoo},
     title = {Product and puzzle formulae for {\(GL_n\)} {Belkale-Kumar} coefficients},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/563},
     zbl = {1232.05239},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/563/}
}
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Allen Knutson; Kevin Purbhoo. Product and puzzle formulae for \(GL_n\) Belkale-Kumar coefficients. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/563

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