Group representation theory MOC

Orthonormality of unitary irreducible representations

Let be concrete realizations of irreps for each . Then with , form an Orthonormal basis of the Group ring under a certain inner product.¹ rep In particular

Should be changed

I think its more productive to view these as elements of

Proof of orthonormality

Let be an arbitrary linear map. Define a linear map

It follows that for any

so is an intertwining operator and therefore by Schur’s lemma either or , so with . Combining these two possibilities gives

If we chose to then , thus

thus the matrix elements fulfil the orthonormality condition.

Proof of spanning set

Let , so we must prove . The tensor product of irreps is reducible, which for concrete reälizations may be stated as

for some numbers . Thus the point-wise product of basic functions is in , and by distributivity is closed under point-wise multiplication. Since Irreps collectively distinguish group elements, it follows from Finite version that . From this and above, is an orthonormal basis under .

Since the number of basis elements equals the dimension of the vector space, it follows that Square sum of irrep dimensions is given by

See also Orthonormality of irreducible characters

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Footnotes

1. 1996, Representations of finite and compact groups, §III.1 ↩