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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.
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.
The Schur multiplier of G is the kernel of any Schur cover and has many interpretations. When the homomorphism is understood, the group D is often called the Schur cover or Darstellungsgruppe. The Schur covers of the symmetric and alternating groups were classified in . The symmetric group of degree n ≥ 4 has Schur covers of order 2⋅n!
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The complex Schur decomposition reads as follows: if A is an n × n square matrix with complex entries, then A can be expressed as [1] [2] [3] = for some unitary matrix Q (so that the inverse Q −1 is also the conjugate transpose Q* of Q), and some upper triangular matrix U. This is called a Schur form of A.