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Schur orthogonality relation
Redirected from "Schur inner product".
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Contents
Idea
Given a finite group of cardinality , then the complex valued class functions have an inner product – the Schur inner product – given by
The Schur orthogonality relation is the statement that the complex irreducible group characters form an orthonormal basis of the space of class functions under this inner product:
(1)
(e.g. Fulton & Harris 91, Thm. 2.12, tom Dieck 09, Prop. 2.2.1 & (2.3.3))
In fact, this holds also at the level of complex irreducible representations (“grand Schur orthogonality”), regarded for each as unitary matrices with matrix entries :
(2)
(e.g. tom Dieck 09 (2.2.7))
Conversely, the sum over all (isomorphism classes) of irreducible characters yields:
(e.g. Fulton & Harris 91, Ex. 2.21, tom Dieck 09, Prop. 2.2.1 & (2.3.3))
References
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Last revised on May 8, 2021 at 11:15:39.
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