Representation Theory of Finite Groups - Applying Schur's Lemma

Applying Schur's Lemma

Lemma. If f: A ⊗ B → C is a morphism of representations, then the corresponding linear transformation obtained by dualizing B is: f′: A → C ⊗ B* is also a morphism of representations. Similarly, if g: A → B ⊗C is a morphism of representations, dualizing it will give another morphism of representations g′: A ⊗ C* → B.

If ρ is an n-dimensional irreducible representation of G with the underlying vector space V, then we can define a G×G morphism of representations, for all g in G and x in V

f: C ⊗ (1_G ⊗ V) → (V ⊗ 1_G)

f:(g ⊗ x) = ρ(g)

where 1_G is the trivial representation of G. This defines a G×G morphism of representations.

Now we use the above lemma and obtain the G×G morphism of representations

.

The dual representation of C as a G×G-representation is equivalent to C. An isomorphism is given if we define the contraction 〈g,h〉 = δ_gh. So, we end up with a G×G-morphism of representations

.

Then

for all x in and y in V.

By Schur's lemma, the image of f″ is a G×G irreducible representation, which is therefore n×n dimensional, which also happens to be a subrepresentation of C (f″ is nonzero).

This is n direct sum equivalent copies V. Note that if ρ₁ and ρ₂ are equivalent G-irreducible representations, the respective images of the intertwining matrices would give rise to the same G×G-irreducible representation of C.

Here, we use the fact that if f is a function over G, then

We convert C into a Hilbert space by introducing the norm where 〈g,h〉 is 1 if g is h and zero otherwise. This is different from the 'contraction' given a couple of paragraphs back, in that this form is sesquilinear. This makes C a unitary representation of G×G. In particular, we now have the concepts of orthogonal complement and orthogonality of subrepresentations.

In particular, if C contains two inequivalent irreducible G×G subrepresentations, then both subrepresentations are orthogonal to each other. To see this, note that for every subspace of a Hilbert space, there exists a unique linear transformation from the Hilbert space to itself which maps points on the subspace to itself while mapping points on its orthogonal complement to zero. This is called the projection map. The projection map associated with the first irreducible representation is an intertwiner. Restricted to the second irreducible representation, it gives an intertwiner from the second irreducible representation to the first. Using Schur's lemma, this must be zero.

Now suppose A ⊗ B is a G×G-irreducible representation of C.

Note. The complex irreducible representations of G×H are always a direct product of a complex irreducible representation of G and a complex irreducible representation of H. This is not the case for real irreducible representations. As an example, there is a 2 dimensional real irreducible representation of the group C₃ × C₃ which transforms nontrivially under both copies of C₃ but cannot be expressed as the direct product of two irreducible representations of C₃.

This representation is also a G-representation (n_A direct sum copies of B where n_A is the dimension of A). If Y is an element of this representation (and hence also of C) and X an element of its dual representation (which is a subrepresentation of the dual representation of C), then

where e is the identity of G. Though the f″ defined a couple of paragraphs back is only defined for G-irreducible representations, and though A ⊗ B is not a G-irreducible representation in general, we claim this argument could be made correct since A ⊗ B is simply the direct sum of copies of Bs, and we have shown that each copy all maps to the same G×G-irreducible subrepresentation of C, we have just showed that as an irreducible G×G-subrepresentation of C is contained in A ⊗ B as another irreducible G×G-subrepresentation of C. Using Schur's lemma again, this means both irreducible representations are the same.

Putting all of this together,

Theorem. C ≅ where the sum is taken over the inequivalent G-irreducible representations V.

Corollary. If there are p inequivalent G-irreducible representations, V_i, each of dimension n_i, then |G| = n₁2 + ... + n_p2.