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Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.\n  However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\\sigma$ is at least an upper bound on the actual class $\\tau$, in the sense that $\\sigma - \\tau$ is a n"},"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":"1410.7419","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-10-08T19:46:56Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"0f50e69b055a74858a36a7de21d608d29b27b61bcf496f92d468ba5bf1bcc339","abstract_canon_sha256":"2693bc4b4cd5bfc53da5d3bc9c13f037c17a4f57bba56ed692a3693cbe0cdc12"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:10:04.639027Z","signature_b64":"pNYY7qfcGpaQRRVAIPCx3U+RAFyGLISaBiFui+P91l0bgSgyywTP7Du3wQCMGR5ar29hXnN7xouQty+UHHuQDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eff2d7be72d0bd3a8f19b5b922b4081d10e3375309603f6916098705b08b2be7","last_reissued_at":"2026-05-18T00:10:04.638344Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:10:04.638344Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cohomology classes of interval positroid varieties and a conjecture of Liu","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.CO","authors_text":"Brendan Pawlowski","submitted_at":"2014-10-08T19:46:56Z","abstract_excerpt":"To each finite subset of $\\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. 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