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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) .
Schur's lemma is frequently applied in the following particular case. Suppose that R is an algebra over a field k and the vector space M = N is a simple module of R. Then Schur's lemma says that the endomorphism ring of the module M is a division algebra over k. If M is finite-dimensional, this division algebra is finite-dimensional.
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.
They satisfy Schur orthogonality relations. The character of a representation ρ is a sum of the matrix coefficients f v i ,η i , where { v i } form a basis in the representation space of ρ, and {η i } form the dual basis .
Around the same time, Dipper and James [6] introduced the quantized Schur algebras (or q-Schur algebras for short), which are a type of q-deformation of the classical Schur algebras described above, in which the symmetric group is replaced by the corresponding Hecke algebra and the general linear group by an appropriate quantum group.
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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]