{"paper":{"title":"Equivariant multiplicities of Coxeter arrangements and invariant bases","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.CO","authors_text":"Atsushi Wakamiko, Hiroaki Terao, Takuro Abe","submitted_at":"2010-11-01T14:25:13Z","abstract_excerpt":"Let $\\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\\A$. A multiplicity $\\bfm : \\A\\rightarrow \\Z$ is said to be equivariant when $\\bfm$ is constant on each $W$-orbit of $\\A$. In this article, we prove that the multi-derivation module $D(\\A, \\bfm)$ is a free module whenever $\\bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of \\cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(\\A, \\bfm)^{W}$ for any multiplicity $\\bfm$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.0329","kind":"arxiv","version":3},"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"}