{"paper":{"title":"Decomposing Inversion Sets of Permutations and Applications to Faces of the Littlewood-Richardson Cone","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.CO","authors_text":"A. McCabe, D. Wehlau, I. Dimitrov, J. Wilson, M. Roth, R. Dewji","submitted_at":"2011-10-26T19:08:46Z","abstract_excerpt":"If $\\alpha \\in S_n$ is a permutation of $\\{1, 2, \\ldots, n\\}$, the inversion set of $\\alpha$ is $\\Phi(\\alpha) = \\{(i, j) \\, | \\, 1 \\leq i < j \\leq n, \\alpha(i) > \\alpha(j)\\}$. We describe all $r$-tuples $\\alpha_1, \\alpha_2, \\ldots, \\alpha_r \\in S_n$ such that $\\Delta_n^+ = \\{(i, j) \\, | \\, 1 \\leq i < j \\leq n\\}$ is the disjoint union of $\\Phi(\\alpha_1), \\Phi(\\alpha_2), \\ldots, \\Phi(\\alpha_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also discuss analogous prob"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.5880","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}