{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T09:53:42Z","timestamp":1772272422171,"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>We define the hive ring, which has a basis indexed by dominant weights for $GL_n({\\Bbb C})$, and structure constants given by counting hives [Knutson-Tao, \"The honeycomb model of $GL_n$ tensor products\"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, \"Involutions on Gel$'$fand-Tsetlin schemes$\\dots$ \"]). We use the octahedron rule from [Robbins-Rumsey, \"Determinants$\\dots$\"] to prove bijectively that this \"ring\" is indeed associative. This, and the Pieri rule, give a self-contained proof that  the hive ring is isomorphic as a ring-with-basis to the representation ring of $GL_n({\\Bbb C})$. In the honeycomb interpretation, the octahedron rule becomes \"scattering\" of the honeycombs. This recovers some of the \"crosses and wrenches\" diagrams from Speyer's  very recent preprint [\"Perfect matchings$\\dots$\"], whose results we use to give a closed form for the associativity bijection.<\/jats:p>","DOI":"10.37236\/1814","type":"journal-article","created":{"date-parts":[[2020,1,11]],"date-time":"2020-01-11T02:32:11Z","timestamp":1578709931000},"source":"Crossref","is-referenced-by-count":24,"title":["A Positive Proof of the Littlewood-Richardson Rule  using the Octahedron Recurrence"],"prefix":"10.37236","volume":"11","author":[{"given":"Allen","family":"Knutson","sequence":"first","affiliation":[]},{"given":"Terence","family":"Tao","sequence":"additional","affiliation":[]},{"given":"Christopher","family":"Woodward","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2004,9,13]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r61\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v11i1r61\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,18]],"date-time":"2020-01-18T04:54:08Z","timestamp":1579323248000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v11i1r61"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2004,9,13]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2004,1,2]]}},"URL":"https:\/\/doi.org\/10.37236\/1814","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2004,9,13]]},"article-number":"R61"}}