Gelfand–Tsetlin graph
E928276
The Gelfand–Tsetlin graph is a combinatorial structure whose vertices encode interlacing patterns corresponding to representations of unitary groups, organizing the branching of these representations in a graded, graph-theoretic form.
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
Bratteli diagram
ⓘ
combinatorial structure ⓘ graded graph ⓘ infinite graph ⓘ |
| connectedTo |
Gel'fand–Tsetlin formulas for matrix elements
ⓘ
determinantal point processes on interlacing arrays ⓘ spectral measures of random Gelfand–Tsetlin patterns ⓘ |
| edgeDefinition |
an edge joins two patterns if one is obtained from the other by deleting the last row and interlacing holds
ⓘ
edges connect interlacing signatures on consecutive levels ⓘ |
| encodes |
branching of irreducible representations of unitary groups
ⓘ
interlacing patterns of signatures ⓘ |
| generalizes | branching graph of symmetric groups via analogous Young graph ⓘ |
| hasAlternativeName |
GT graph
NERFINISHED
ⓘ
Gelfand–Tsetlin branching graph NERFINISHED ⓘ |
| hasBoundary |
Martin boundary of the graph
ⓘ
Thoma-type boundary describing extreme characters of U(∞) ⓘ |
| hasCombinatorialModel | triangular arrays of integers with interlacing inequalities GENERATED ⓘ |
| hasLevel0Vertex | empty signature at level 0 ⓘ |
| hasLocalFiniteProperty | each vertex has finitely many neighbors on adjacent levels ⓘ |
| hasNaturalOrientation | edges oriented from lower to higher levels ⓘ |
| hasOrigin | introduced in the context of constructing bases for representations of classical groups ⓘ |
| hasPathSpace | infinite paths correspond to coherent systems of measures on signatures ⓘ |
| hasSymmetry | invariance under simultaneous shifts of all coordinates of a signature ⓘ |
| hasVertexSetDescription |
vertices are Gelfand–Tsetlin patterns
ⓘ
vertices encode interlacing integer arrays ⓘ |
| isCountable | vertex set is countable ⓘ |
| isGradedBy | rank n of the unitary group U(n) ⓘ |
| levelStructure | n-th level corresponds to signatures of length n ⓘ |
| mathematicalDiscipline |
asymptotic combinatorics
ⓘ
combinatorics ⓘ probability theory ⓘ representation theory ⓘ |
| namedAfter |
Israel Gelfand
NERFINISHED
ⓘ
Mikhail Tsetlin NERFINISHED ⓘ |
| organizes | branching of representations U(1) ⊂ U(2) ⊂ U(3) ⊂ ⋯ ⓘ |
| relatedTo |
Gelfand–Tsetlin basis
NERFINISHED
ⓘ
Gelfand–Tsetlin patterns NERFINISHED ⓘ Young graph NERFINISHED ⓘ branching graph of U(∞) ⓘ representation theory of classical groups ⓘ representation theory of unitary groups ⓘ |
| studiedIn |
asymptotic representation theory
ⓘ
probability on combinatorial structures ⓘ random matrix theory ⓘ |
| usedFor |
constructing probability measures on paths corresponding to characters
ⓘ
describing inductive limits of unitary groups ⓘ parametrizing irreducible characters of U(∞) ⓘ studying harmonic analysis on infinite-dimensional unitary groups ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.