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Denote by $\\Fo = G/B$ the flag variety, by $X_w$ the Schubert subvariety of $\\Fo$ associated with an element $w\\in W$, and by $C_w$ the tangent cone to $X_w$ at the point $p = eB$. Then $C_w$ is a subscheme of the tangent space $T_pX_w\\subseteq T_p\\Fo$. Suppose $w$, $w'$ are distinct involutions in $W$. Using the so-called Kostant--Kumar polynomials, we show that if every irreducible component of $\\Phi$ i"},"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":"1210.5740","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-10-21T17:36:33Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"c56b6e3b019543e0ec7db53ed935c0d4ebe94506760f5d6d2a2e3b4f08610a29","abstract_canon_sha256":"925ce552d66179dad4df6363951135c47680262c615f8b9fc98033caf7eff62b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:03.528045Z","signature_b64":"EuNF78iVVZcbp5ux1vVCUgpUIoEMZ+Y+S7UsBn9oBjIaoNqRhWHSJcZJcGDJUYAXOC7dcevVVoX8DEbQp5G0BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27f42c5ea3b873eaa8b1be765881584fcffa724101a287e5f2c6378656820378","last_reissued_at":"2026-05-18T02:40:03.527612Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:03.527612Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Kostant--Kumar polynomials and tangent cones to Schubert varieties for involutions in $A_n$, $F_4$ and $G_2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.RT","authors_text":"Dmitriy Y. Eliseev, Mikhail V. Ignatyev","submitted_at":"2012-10-21T17:36:33Z","abstract_excerpt":"Let $G$ be a reductive complex algebraic group, $T$ a maximal torus of $G$, $B$ a Borel subgroup of $G$ containing $T$, $\\Phi$ the root system of $G$ w.r.t. $T$, $W$ the Weyl group of $\\Phi$. Denote by $\\Fo = G/B$ the flag variety, by $X_w$ the Schubert subvariety of $\\Fo$ associated with an element $w\\in W$, and by $C_w$ the tangent cone to $X_w$ at the point $p = eB$. Then $C_w$ is a subscheme of the tangent space $T_pX_w\\subseteq T_p\\Fo$. Suppose $w$, $w'$ are distinct involutions in $W$. Using the so-called Kostant--Kumar polynomials, we show that if every irreducible component of $\\Phi$ i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.5740","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1210.5740","created_at":"2026-05-18T02:40:03.527670+00:00"},{"alias_kind":"arxiv_version","alias_value":"1210.5740v1","created_at":"2026-05-18T02:40:03.527670+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.5740","created_at":"2026-05-18T02:40:03.527670+00:00"},{"alias_kind":"pith_short_12","alias_value":"E72CYXVDXBZ6","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_16","alias_value":"E72CYXVDXBZ6VKFR","created_at":"2026-05-18T12:27:04.183437+00:00"},{"alias_kind":"pith_short_8","alias_value":"E72CYXVD","created_at":"2026-05-18T12:27:04.183437+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7","json":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7.json","graph_json":"https://pith.science/api/pith-number/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/graph.json","events_json":"https://pith.science/api/pith-number/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/events.json","paper":"https://pith.science/paper/E72CYXVD"},"agent_actions":{"view_html":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7","download_json":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7.json","view_paper":"https://pith.science/paper/E72CYXVD","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1210.5740&json=true","fetch_graph":"https://pith.science/api/pith-number/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/graph.json","fetch_events":"https://pith.science/api/pith-number/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/action/storage_attestation","attest_author":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/action/author_attestation","sign_citation":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/action/citation_signature","submit_replication":"https://pith.science/pith/E72CYXVDXBZ6VKFRXZ3FRAKYJ7/action/replication_record"}},"created_at":"2026-05-18T02:40:03.527670+00:00","updated_at":"2026-05-18T02:40:03.527670+00:00"}