Gelfand–Tsetlin basis

E270384

basis in representation theory canonical basis combinatorial basis

The Gelfand–Tsetlin basis is a canonical, combinatorially defined basis for representations of certain Lie algebras and groups, particularly used in the representation theory of GL(n) and related structures.

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All labels observed (4)

Label	Occurrences
Gelfand–Tsetlin patterns	2
Gelfand–Tsetlin basis canonical	1
Gelfand–Tsetlin integrable system	1
Gelfand–Tsetlin tableaux	1

Statements (48)

Predicate	Object
instanceOf	basis in representation theory ⓘ canonical basis ⓘ combinatorial basis ⓘ
appliesTo	polynomial representations of GL(n) ⓘ representations with integral highest weights ⓘ
associatedWith	Gelfand–Tsetlin graph ⓘ Gelfand–Tsetlin basis self-linksurface differs ⓘ surface form: Gelfand–Tsetlin patterns branching rules for GL(n) ⓘ branching rules for unitary groups ⓘ
constructedAlong	chain gl(1) ⊂ gl(2) ⊂ … ⊂ gl(n) ⓘ chain of groups GL(1) ⊂ GL(2) ⊂ … ⊂ GL(n) ⓘ
definedFor	finite-dimensional irreducible representations of GL(n,ℂ) ⓘ finite-dimensional irreducible representations of gl(n,ℂ) ⓘ highest weight representations ⓘ
developedIn	mid 20th century ⓘ
field	mathematics ⓘ
gives	simultaneous eigenbasis for a maximal commutative subalgebra of U(gl(n)) ⓘ weight basis for representations of gl(n) ⓘ
hasConstructionMethod	combinatorial construction ⓘ inductive construction along a chain of subalgebras ⓘ
hasProperty	canonical up to normalization ⓘ compatible with restriction along the chain GL(1) ⊂ … ⊂ GL(n) ⓘ elements indexed by integer arrays satisfying interlacing inequalities ⓘ
namedAfter	Israel Gelfand ⓘ Mikhail Tsetlin ⓘ
parameterizedBy	Gelfand–Tsetlin basis self-linksurface differs ⓘ surface form: Gelfand–Tsetlin patterns Gelfand–Tsetlin basis self-linksurface differs ⓘ surface form: Gelfand–Tsetlin tableaux triangular arrays of integers or half-integers ⓘ
relatedTo	Gelfand–Tsetlin algebra ⓘ Gelfand–Tsetlin basis self-linksurface differs ⓘ surface form: Gelfand–Tsetlin integrable system Young diagrams ⓘ crystal bases ⓘ highest weight theory ⓘ
satisfies	interlacing conditions between rows of patterns ⓘ
subfield	Lie theory ⓘ representation theory ⓘ
usedFor	construction of Gelfand–Tsetlin integrable systems ⓘ explicit computation of Clebsch–Gordan coefficients ⓘ explicit computation of matrix elements ⓘ explicit description of representation branching ⓘ spectral analysis of commuting operators ⓘ
usedIn	representation theory ⓘ representation theory of GL(n) ⓘ representation theory of Lie algebras ⓘ representation theory of Lie groups ⓘ representation theory of classical Lie algebras ⓘ representation theory of general linear groups ⓘ representation theory of unitary groups ⓘ

Referenced by (5)

Full triples — surface form annotated when it differs from this entity's canonical label.

Israel Gelfand → knownFor → Gelfand–Tsetlin basis ⓘ

Gelfand–Tsetlin basis → parameterizedBy → Gelfand–Tsetlin basis self-linksurface differs ⓘ

this entity surface form: Gelfand–Tsetlin patterns

Gelfand–Tsetlin basis → parameterizedBy → Gelfand–Tsetlin basis self-linksurface differs ⓘ

this entity surface form: Gelfand–Tsetlin tableaux

Gelfand–Tsetlin basis → associatedWith → Gelfand–Tsetlin basis self-linksurface differs ⓘ

this entity surface form: Gelfand–Tsetlin patterns

Gelfand–Tsetlin basis → relatedTo → Gelfand–Tsetlin basis self-linksurface differs ⓘ

this entity surface form: Gelfand–Tsetlin integrable system