{"paper":{"title":"Rook sums in the symmetric group algebra","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Darij Grinberg","submitted_at":"2025-07-30T05:16:39Z","abstract_excerpt":"Let $\\mathcal{A}$ be the group algebra $\\mathbf{k}[S_n]$ of the $n$-th symmetric group $S_n$ over a commutative ring $\\mathbf{k}$. For any two subsets $A$ and $B$ of $[n]$, we define the elements \\[ \\nabla_{B,A}:=\\sum_{\\substack{w\\in S_n;\\\\w\\left( A\\right) =B}} w \\qquad \\text{and} \\qquad \\widetilde{\\nabla}_{B,A}:=\\sum_{\\substack{w\\in S_n;\\\\w\\left( A\\right) \\subseteq B}}w \\] of $\\mathcal{A}$. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). We express the product $\\nabla_{D,C}\\nabla_{B,A}$ as a $\\mathbb{Z}$-lin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.22386","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.22386/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}