orthogonality relations

First orthogonality relations: Let ρα:GVα and ρβ:GVβ be irreducible representations of a finite groupMathworldPlanetmath G over the field . Then


We have the following useful corollary. Let χ1, χ2 be charactersPlanetmathPlanetmath of representationsPlanetmathPlanetmath V1, V2 of a finite group G over a field k of characteristicPlanetmathPlanetmath 0. Then


First of all, consider the special case where V=k with the trivial action of the group. Then HomG(k,V2)V2G, the fixed points. On the other hand, consider the map


(with the sum in End(V2)). Clearly, the image of this map is contained in V2G, and it is the identityPlanetmathPlanetmath restricted to V2G. Thus, it is a projectionMathworldPlanetmathPlanetmath with image V2G. Now, the rank of a projection (over a field of characteristic 0) is its trace. Thus,


which is exactly the orthogonality formula for V1=k.

Now, in general, Hom(V1,V2)V1*V2 is a representation, and HomG(V1,v2)=(Hom(V1,V2))G. Since χV1*V2=χ1¯χ2,


which is exactly the relation we desired. ∎

In particular, if V1,V2 irreducible, by Schur’s Lemma


where D is a division algebra. In particular, non-isomorphic irreducible representations have orthogonalMathworldPlanetmath characters. Thus, for any representation V, the multiplicitiesMathworldPlanetmath ni in the unique decomposition of V into the direct sumPlanetmathPlanetmath (http://planetmath.org/DirectSum) of irreducibles


where Vi ranges over irreducible representations of G over k, can be determined in terms of the character inner product:


where ψ is the character of V and χi the character of Vi. 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 k is algebraically closedMathworldPlanetmath, the only finite division algebra over k is k itself, so the characters of irreducible representations form an orthonormal basis for the vector spaceMathworldPlanetmath of class functions with respect to this inner product. Since (χi,χi)=1 for all irreducibles, the multiplicity formula above reduces to ni=(ψ,χi).

Second orthogonality relations: We assume now that k is algebraically closed. Let g,g be elements of a finite group G. Then


where the sum is over the characters of irreducible representations, and CG(g) is the centralizerMathworldPlanetmathPlanetmathPlanetmath of g.


Let χ1,,χn be the characters of the irreducible representations, and let g1,,gn be representatives of the conjugacy classesMathworldPlanetmathPlanetmath.

Let A be the matrix whose ijth entry is |G:CG(gj)|(χi(gj)¯). By first orthogonality, AA*=|G|I (here * denotes conjugate transposeMathworldPlanetmath), where I is the identity matrixMathworldPlanetmath. Since left inversesMathworldPlanetmath (http://planetmath.org/MatrixInverse) are right , A*A=|G|I. Thus,


Replacing gi or gk with any conjuagate will not change the expression above. thus, if our two elements are not conjugatePlanetmathPlanetmathPlanetmath, we obtain that χχ(g)χ(g)¯=0. On the other hand, if gg, then i=k in the sum above, which reduced to the expression we desired. ∎

A special case of this result, applied to 1 is that |G|=χχ(1)2, that is, the sum of the squares of the dimensionsPlanetmathPlanetmath (http://planetmath.org/Dimension) of the irreducible representations of any finite group is the order of the group.

Title orthogonality relations
Canonical name OrthogonalityRelations
Date of creation 2013-03-22 13:21:27
Last modified on 2013-03-22 13:21:27
Owner mhale (572)
Last modified by mhale (572)
Numerical id 16
Author mhale (572)
Entry type Theorem
Classification msc 20C15