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We determine the abelian subgroups $A\\subset G$ for which there is a polar decomposition $G=KAH$, and we determine for which minimal parabolic subgroups $P\\subset G$, the orbit $PH$ is open in $G/H$."},"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":"1301.0489","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2013-01-03T16:02:58Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"33f2bb9de3077ee1ccd91ebc82c407837436dbeb9307d23c590c08c07495c22a","abstract_canon_sha256":"5d0e6f0b618e3862363e299b80dcc6312fc2087e8e09b1e4dd4d9791a3384500"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:48.373693Z","signature_b64":"pZ2Jf8nh992aX4cQthiYGgKYyjLytwrZXPfOQFGPAAp287PGxF449W9kxZWHBSuJnLTV6Ndyo2e4Bx/7GqHrBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb01d5b01f3228f6fa29eef395399774eae751665f8336d2dc9242729bdabc71","last_reissued_at":"2026-05-18T02:28:48.373324Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:48.373324Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposition Theorems for Triple Spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.DG","authors_text":"Bernhard Kr\\\"otz, Henrik Schlichtkrull, Thomas Danielsen","submitted_at":"2013-01-03T16:02:58Z","abstract_excerpt":"A triple space is a homogeneous space $G/H$ where $G=G_0\\times G_0\\times G_0$ is a threefold product group and $H\\simeq G_0$ the diagonal subgroup of $G$. 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