Schur orthogonality relations

In mathematics, the Schur orthogonality relations, which is 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).

The space of complex-valued class functions of a finite group G has a natural inner product:

where ${\displaystyle {\overline {\beta (g)}}}$ means the complex conjugate of the value of ${\displaystyle \beta }$ on g. With respect to this inner product, the irreducible characters form an orthonormal basis for the space of class functions, and this yields the orthogonality relation for the rows of the character table:

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