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We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, $\\mathcal E_G$ is a free $\\mathcal E_H$-module for any $H\\subseteq G$, and if $\\mathcal E_G$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases "},"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":"1310.4112","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.QA","submitted_at":"2013-10-15T16:47:05Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"baa503ceb8d604c1cb8e68621fd8a1ea5b6439fa602c0c66c7676d20772171b0","abstract_canon_sha256":"958bb86aef035552c5416959735583905353cda128a2d33f25072018afcb0ee3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:35.015588Z","signature_b64":"+M6/+L5wkXwrBgYPLAN/jJuyTlHwri0xVAkp6DWU48JET4Pu4eqG5MC6f8YfFOvYAjlW+EnSg1hJy6mPX+FKDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4645dcf468f8c5c30b71dccb5e0dd53a0732d454ddb8c817646f83353094313c","last_reissued_at":"2026-05-18T02:56:35.014819Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:35.014819Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Subalgebras of the Fomin-Kirillov algebra","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.QA","authors_text":"Jonah Blasiak, Karola M\\'esz\\'aros, Ricky Ini Liu","submitted_at":"2013-10-15T16:47:05Z","abstract_excerpt":"The Fomin-Kirillov algebra $\\mathcal E_n$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on $n$ vertices. 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