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We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups"},"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":"1609.02046","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-07T16:07:18Z","cross_cats_sorted":["math.QA","math.RT"],"title_canon_sha256":"10d055cdbd7ea9d21a1befe624c140497084dcc9483bc0145a652216093bf3a9","abstract_canon_sha256":"9bf4ee85a2384c61c0544a72681a1d4db67d7b20ce3b567441289b301586051c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:13.893265Z","signature_b64":"/8/dpOvO473vL9Q1LutPWj2rrbKqVyO44FOEUMlo4JEk6Xu78FUaOhk5XF10Vsa7N8QyLXJ20Pf7DlaXGeXoDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14f8ec1d717564820c0542b08d8d4a8f9cc48417ce0cebc2d2f1eb50dbd2e310","last_reissued_at":"2026-05-18T00:38:13.892628Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:13.892628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Berenstein-Kirillov group and cactus groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.QA","math.RT"],"primary_cat":"math.CO","authors_text":"Max Glick, Michael Chmutov, Pavlo Pylyavskyy","submitted_at":"2016-09-07T16:07:18Z","abstract_excerpt":"Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. 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