Schur–Weyl duality

E503508

mathematical theorem result in representation theory

Schur–Weyl duality is a fundamental result in representation theory that links representations of the symmetric group and the general linear group via their commuting actions on tensor powers of a vector space.

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Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in representation theory ⓘ
appliesTo	tensor power V^{\otimes n} of a vector space V ⓘ
characteristicAssumption	often formulated over fields of characteristic zero ⓘ
connects	Specht modules of the symmetric group ⓘ irreducible polynomial representations of GL(V) ⓘ
context	finite-dimensional vector spaces ⓘ
describes	bimodule structure of V^{\otimes n} ⓘ
extendsTo	fields of positive characteristic with modifications ⓘ
field	algebra ⓘ group theory ⓘ representation theory ⓘ
formalism	bimodule decomposition ⓘ
generalizedBy	Howe duality NERFINISHED ⓘ
gives	decomposition of V^{\otimes n} into irreducible GL(V)-modules and S_n-modules ⓘ
hasVariant	Schur–Weyl duality for Hecke algebras NERFINISHED ⓘ Schur–Weyl duality for quantum groups NERFINISHED ⓘ q-Schur–Weyl duality ⓘ
historicalPeriod	early 20th century ⓘ
implies	V^{\otimes n} \cong \bigoplus_{\lambda} S^{\lambda}(V) \otimes M^{\lambda} ⓘ double centralizer property for GL(V) and S_n ⓘ
involves	group algebra of the symmetric group ⓘ representation of GL(V) ⓘ representation of S_n ⓘ
namedAfter	Hermann Weyl NERFINISHED ⓘ Issai Schur NERFINISHED ⓘ
relates	centralizer algebra of GL(V) on V^{\otimes n} ⓘ centralizer algebra of S_n on V^{\otimes n} ⓘ general linear group NERFINISHED ⓘ symmetric group ⓘ tensor powers of a vector space ⓘ
requires	dimension of V at least n for full correspondence ⓘ
statesThat	actions of GL(V) and S_n on V^{\otimes n} commute ⓘ images of GL(V) and S_n actions on V^{\otimes n} are mutual centralizers ⓘ
typeOf	duality between groups and centralizer algebras ⓘ
usedIn	Schur–Weyl reciprocity NERFINISHED ⓘ algebraic combinatorics ⓘ categorification ⓘ construction of Schur algebras ⓘ invariant theory ⓘ representation theory of GL_n ⓘ representation theory of S_n ⓘ theory of symmetric functions ⓘ
uses	Schur functors NERFINISHED ⓘ Young diagrams NERFINISHED ⓘ commuting group actions ⓘ partitions of n ⓘ

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The Classical Groups: Their Invariants and Representations → topic → Schur–Weyl duality ⓘ