{"paper":{"title":"Enumeration of $\\C{H}$-strata in quantum matrices with respect to dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.QA","authors_text":"Jason Bell, Karel Casteels, St\\'ephane Launois","submitted_at":"2010-09-13T18:32:17Z","abstract_excerpt":"We present a combinatorial method to determine the dimension of $\\C{H}$-strata in the algebra of $m\\times n$ quantum matrices $\\Oq$ as follows. To a given $\\C{H}$-stratum we associate a certain permutation via the notion of pipe-dreams. We show that the dimension of the $\\C{H}$-stratum is precisely the number of odd cycles in this permutation. Using this result, we are able to give closed formulas for the trivariate generating function that counts the $d$-dimensional $\\C{H}$-strata in $\\Oq$. Finally, we extract the coefficients of this generating function in order to settle conjectures propose"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1009.2474","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"}