First of all, consider the special case where with the trivial action of the group. Then , the fixed points. On the other hand, consider the map
(with the sum in ). Clearly, the image of this map is contained in , and it is the identity restricted to . Thus, it is a projection with image . Now, the rank of a projection (over a field of characteristic 0) is its trace. Thus,
which is exactly the orthogonality formula for .
Now, in general, is a representation, and . Since ,
which is exactly the relation we desired. ∎
In particular, if irreducible, by Schur’s Lemma
where is a division algebra. In particular, non-isomorphic irreducible representations have orthogonal characters. Thus, for any representation , the multiplicities in the unique decomposition of into the direct sum (http://planetmath.org/DirectSum) of irreducibles
where ranges over irreducible representations of over , can be determined in terms of the character inner product:
where is the character of and the character of . In particular, representations over a field of characteristic zero are determined by their character. Note: This is not true over fields of positive characteristic.
If the field is algebraically closed, the only finite division algebra over is itself, so the characters of irreducible representations form an orthonormal basis for the vector space of class functions with respect to this inner product. Since for all irreducibles, the multiplicity formula above reduces to .
Second orthogonality relations: We assume now that is algebraically closed. Let be elements of a finite group . Then
where the sum is over the characters of irreducible representations, and is the centralizer of .
Let be the characters of the irreducible representations, and let be representatives of the conjugacy classes.
Let be the matrix whose th entry is . By first orthogonality, (here denotes conjugate transpose), where is the identity matrix. Since left inverses (http://planetmath.org/MatrixInverse) are right , . Thus,
Replacing or with any conjuagate will not change the expression above. thus, if our two elements are not conjugate, we obtain that . On the other hand, if , then in the sum above, which reduced to the expression we desired. ∎
A special case of this result, applied to is that , that is, the sum of the squares of the dimensions (http://planetmath.org/Dimension) of the irreducible representations of any finite group is the order of the group.
|Date of creation||2013-03-22 13:21:27|
|Last modified on||2013-03-22 13:21:27|
|Last modified by||mhale (572)|