{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,6]],"date-time":"2026-04-06T09:12:22Z","timestamp":1775466742529,"version":"3.50.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 $2$.<\/jats:p>","DOI":"10.37236\/563","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T22:51:43Z","timestamp":1578696703000},"source":"Crossref","is-referenced-by-count":9,"title":["Product and Puzzle Formulae for $GL_n$ Belkale-Kumar Coefficients"],"prefix":"10.37236","volume":"18","author":[{"given":"Allen","family":"Knutson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kevin","family":"Purbhoo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2011,3,31]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p76\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v18i1p76\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T18:14:37Z","timestamp":1579284877000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v18i1p76"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3,31]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2011,1,5]]}},"URL":"https:\/\/doi.org\/10.37236\/563","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,3,31]]},"article-number":"P76"}}