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.
Read more about this topic: Character Table
Famous quotes containing the word relations:
“Happy will that house be in which the relations are formed from character; after the highest, and not after the lowest order; the house in which character marries, and not confusion and a miscellany of unavowable motives.”
—Ralph Waldo Emerson (1803–1882)