{"paper":{"title":"Mullineux map: $d$-balanced partitions and $d$-runner matrices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Pavel Turek","submitted_at":"2025-04-04T18:40:16Z","abstract_excerpt":"Let $d,e>1$ be two integers. For $e$ prime, the Mullineux map $m_e$ describes tensor products of the irreducible modules of symmetric groups with the sign in characteristic $e$ as well as certain entries of decomposition matrices. Motivated by understanding new columns of decomposition matrices, we prove that if $\\lambda$ is an $e$-regular partition such that $d$ divides the arm length of any rim hook of $\\lambda$ of size divisible by $e$, then $m_e(\\lambda)'$ is a partition such that the arm length of any of its rim hooks of size divisible by $e$ is congruent to $-1$ modulo $d$. We introduce "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.03864","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.03864/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"}