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In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups , such as the rotation group SO(3) .
In mathematics, Schur's lemma [1] is an elementary but extremely useful statement in representation theory of groups and algebras.In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then either φ is invertible, or φ = 0.
In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map from the endomorphisms of V ...
Schur 1. Issai Schur Issai Schur 2. Schur's lemma states that a G-linear map between irreducible representations must be either bijective or zero. 3. The Schur orthogonality relations on a compact group says the characters of non-isomorphic irreducible representations are orthogonal to each other. 4.
Schur algebra; Schur complement; Schur decomposition; Schur functor; Schur multiplier; Schur orthogonality relations; Schur polynomial; Schur product theorem; Schur test; Schur–Weyl duality; Schur–Zassenhaus theorem; Issai Schur; Schur's inequality; Schur's lemma; Schur's property
Standard representation theory for finite groups has a square character table with row and column orthogonality properties. With a slightly different definition of conjugacy classes and use of the intertwining number, a square character table with similar orthogonality properties also exists for the corepresentations of finite magnetic groups. [2]
In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. Green first published in ...
The orthogonality relations can aid many computations including: Decomposing an unknown character as a linear combination of irreducible characters. Constructing the complete character table when only some of the irreducible characters are known. Finding the orders of the centralizers of representatives of the conjugacy classes of a group.