Character Table - Orthogonality Relations

Orthogonality Relations

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

where means the complex conjugate of the value of 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:

For the orthogonality relation for columns is as follows:

where the sum is over all of the irreducible characters of G and the symbol denotes the order of the centralizer of .

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.
• Finding the order of the group.