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The purpose of this paper is to prove that all Frobenius eigenvalues of the $i$-th cohomology $\\mathcal{H}^i(\\mathcal{L})|_x$ for a simple perverse sheaf $\\mathcal{L}\\in \\mathcal{Q}_{\\alpha}$ and $x\\in \\mathbb{E}_{\\alpha}^{F^n}=\\mathbb{E}_{\\alpha}(\\mathbb{F}_{q^n})$ are equal to $(\\sqrt{q^n})^{i}$ as a conjecture given by Schiffmann (\\cite{Schiffmann2}). As an application, we prove the existence of a class of Hall polynomials."},"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":"1712.04167","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2017-12-12T08:24:01Z","cross_cats_sorted":["math.AG","math.QA"],"title_canon_sha256":"9c5c295cf68cea1bb1872c2ba3d22ba3994f92d9e0ea7b76a5bb6bda2cec11eb","abstract_canon_sha256":"386548c5c54544696729deafa50de5fae5553b4ade643dd5dc174cf7fce556e7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:12.012256Z","signature_b64":"GMHqucGIYz5m4qb3GTduRyD2EMdYpK2IoL/0Knp/we1V69Ps3/o10gcW/x7veILdZ33sS34d5Ogrqvy5OUK7DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cca7ebf46e48e252914e8b059c5bb9791cea613694e6cb54c8909776f84d5b25","last_reissued_at":"2026-05-18T00:06:12.011742Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:12.011742Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On purity theorem of Lusztig's perverse sheaves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.QA"],"primary_cat":"math.RT","authors_text":"Fan Xu, Jie Xiao, Minghui Zhao","submitted_at":"2017-12-12T08:24:01Z","abstract_excerpt":"Let $Q$ be a finite quiver without loops and $\\mathcal{Q}_{\\alpha}$ be the Lusztig category for any dimension vector $\\alpha$. 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