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As an application, we introduce the factorial $G\\Theta / G\\Theta'$-functions representing the torus equivariant $K$-theoretic Schubert classes of the symplectic and the odd orthogonal Grassmannians, which generalize the (double) theta polynomials of Buch--"},"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":"1504.02828","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-04-11T03:41:29Z","cross_cats_sorted":["math.CO","math.RT"],"title_canon_sha256":"85b7e809ea93c47ff1a671d6f9ef7cdf44288b38b731ab79ec11e57ceb49243e","abstract_canon_sha256":"b993cc457c5e92f664b148a0cde3a20a0b0a33adc92db12066c4f457563fe1ec"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:42.202241Z","signature_b64":"mulU77Z2Kqnl6PGD9Swgmw45ERAwah/MOcTCeTmAWF0IZvZAUZmlUlbtMatJk0y7KMbzIli8zuu00wXrScpXBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66f5a8e140c0dfc5a100b9f36bf640ca51fd6bd5fbe5570845cdba497d0f444f","last_reissued_at":"2026-05-18T00:04:42.201598Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:42.201598Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Degeneracy Loci Classes in $K$-theory - Determinantal and Pfaffian Formula -","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.RT"],"primary_cat":"math.AG","authors_text":"Hiroshi Naruse, Takeshi Ikeda, Thomas Hudson, Tomoo Matsumura","submitted_at":"2015-04-11T03:41:29Z","abstract_excerpt":"We prove a determinantal formula and Pfaffian formulas that respectively describe the $K$-theoretic degeneracy loci classes for Grassmann bundles and for symplectic Grassmann and odd orthogonal bundles. 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