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Denote by $W(e)$ and $W_*(e)$ the centralizer and stabilizer of $e\\in \\Lambda$ in $W$, respectively. Let $W(e)$ act by conjugation on the set of left cosets of $W_*(e)$ in $W$. We find that $ue$ and $ve$ ($u, v\\in W$) are conjugate if and only if $uW_*(e)$ and $vW_*(e)$ are in the same orbit. As consequences, there is a one-to-one correspondence between the co"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1205.5480","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-05-24T15:18:39Z","cross_cats_sorted":[],"title_canon_sha256":"ccb8cb43f7dc872ac3dd5de0092d9577852cf34aacd96bacbcfcb8c44284fcb1","abstract_canon_sha256":"db43d54b7a741ffb72b1b57ecd515ecd0afa5104e465baae67f027386fd6b6e8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:06.792649Z","signature_b64":"y/BVXgRS4FwV8trsXOTbsYP4eeLpTm5ARv11zUi3FxpJNPP61IMLmNHCDdFdgtD91PYvsuzA3L2sSVZAOuFNBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"39f1bee0197840741e026b572ad59895c037ef0f908ea5821a71549b8c4b50b9","last_reissued_at":"2026-05-18T03:46:06.792010Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:06.792010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Conjugacy Classes of Renner Monoids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RT","authors_text":"You'an Cao, Zhenheng Li, Zhuo Li","submitted_at":"2012-05-24T15:18:39Z","abstract_excerpt":"In this paper we describe conjugacy classes of a Renner monoid $R$ with unit group $W$, the Weyl group. We show that every element in $R$ is conjugate to an element $ue$ where $u\\in W$ and $e$ is an idempotent in a cross section lattice. Denote by $W(e)$ and $W_*(e)$ the centralizer and stabilizer of $e\\in \\Lambda$ in $W$, respectively. Let $W(e)$ act by conjugation on the set of left cosets of $W_*(e)$ in $W$. We find that $ue$ and $ve$ ($u, v\\in W$) are conjugate if and only if $uW_*(e)$ and $vW_*(e)$ are in the same orbit. 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