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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.
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]
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Several expressions arise for this relation, one of the most important being the expansion of the Schur functions s λ in terms of the symmetric power functions =. If we write χ λ ρ for the character of the representation of the symmetric group indexed by the partition λ evaluated at elements of cycle type indexed by the partition ρ, then