Math 202C: May 20 Lecture

*** Problems in this lecture due 05/24/2026 at 23:59 ***

Let G be an arbitrary group, and let (V,\varphi) be a finite-dimensional unitary representation of G. Then, there exists an orthogonal direct sum decomposition

V = U_1 \oplus \dots \oplus U_k

of V into nonzero G-invariant and G-irrreducible subspaces. This fact requires nothing from the group G, what matters is that the representation space V is a finite-dimensional Hilbert space and the action \varphi represents G as a group of unitary transformations.

A better way to organize this decomposition is to introduce a set \Lambda parameterizing isomorphism classes of irreducible finite-dimensional unitary representations of G. Since we are making no assumptions on G, nothing more specific can be said about the parameter set \Lambda (as we know from Math 202B, if G is a finite group then \Lambda is a finite set whose cardinality is equal to the number of conjugacy classes in G, but we are not assuming that here). For each \lambda \in \Lambda, fix a representative (U^\lambda,\rho^\lambda) of the corresponding isomorphism class of irreducible finite-dimensional unitary representations of G. Then, we can organize the above decomposition of (V,\varphi) as

V= \bigoplus\limits_{\lambda \in \Lambda} V^\lambda,

where each constituent of this orthogonal (internal) direct sum further decomposes as

V^\lambda = U_1^\lambda \oplus \dots \oplus U^\lambda_{m^\lambda(V)},

with each U_i^\lambda isomorphic to the irrep U^\lambda. We allow the possibility m^\lambda(V)=0, which means precisely that V^\lambda = \{0_V\}. What we are doing here is building the same binary tree of pairs of complementary subspaces that produces the decomposition V=U_1 \oplus \dots \oplus \dots U_k, and then grouping the leaves of the tree corresponding to their isomorphism type \lambda \in \Lambda. This means that the decomposition

V = \bigoplus\limits_{\lambda \in \Lambda} V^\lambda

into isotypic components is canonical, but the decomposition of each isotypic component V^\lambda itself is not. What we can say is that the isotypic decomposition of V can instead be written as the external direct sum

V \simeq \bigoplus\limits_{\lambda \in \Lambda} m^\lambda(V)U^\lambda,

where this is not an equality but an isomorphism of representations

Lately we have argued that a still better way to think about this is to replace the numerical multiplicity m^\lambda(V) with a multiplicity space: we write the isotypic decomposition of (V,\varphi) as

V \simeq \bigoplus\limits_{\lambda \in \Lambda} M^\lambda(V) \otimes U^\lambda

with M^\lambda(V) = \mathrm{Hom}_G(U^\lambda,V). Thanks to Schur’s Lemma, each nonzero linear transformation A \in \mathrm{Hom}_G(U^\lambda,V) is an injection of U^\lambda into V, so we can think of the multiplicity space M^\lambda(V) as parameterizing all ways in which the irreducible representation U^\lambda can be realized inside the representation V in a G-equivariant way. In particular, there is an evaluation map

M^\lambda(V) \otimes U^\lambda \longrightarrow V^\lambda

given by A \otimes u \mapsto Au. To make this spacial (as opposed to numerical) version of the isotypic decomposition legitimate, we still have to say how each of its components M^\lambda(V) \otimes U^\lambda is being viewed as a unitary representation of G. This means that M^\lambda(V) \otimes U^\lambda must be a Hilbert space on which G acts by linear isometries. Concerning the Hilbert space structure on M^\lambda(V) \otimes U^\lambda, we have discussed in lecture how to make a tensor product of Hilbert spaces into a Hilbert space in lecture, and also U^\lambda is by definition a Hilbert space, so all that is left here is to define a scalar product on M^\lambda(V)=\mathrm{Hom}_G(U^\lambda,V). To do this, we simply take the Frobenius-Hilbert-Schmidt scalar product on \mathrm{Hom}(U^\lambda,V), in which M^\lambda(V) sits as a subspace. Now for the action of G on the Hilbert space M^\lambda(V) \otimes U^\lambda we take this simplest possible recipe, namely

g(A \otimes u) = A \otimes \varphi(g)u.

Now we want to write down a proof of our proposed spacial reformulation

V \simeq \bigoplus\limits_{\lambda \in \Lambda} M^\lambda(V) \otimes U^\lambda

of the numerical isotypic decomposition

V \simeq \bigoplus\limits_{\lambda \in \Lambda} m^\lambda(V)U^\lambda.

This means that we should give an isomormphism of representations

\bigoplus\limits_{\lambda \in \Lambda} M^\lambda(V) \otimes U^\lambda \simeq \bigoplus\limits_{\lambda \in \Lambda} m^\lambda(V)U^\lambda.

Problem 1: Reduce the above to giving an isomorphism of representations M^\lambda(V) \otimes U^\lambda \simeq m^\lambda(V)U^\lambda for each \lambda \in \Lambda, and give the required isomorphism.

Once Problem 1 is complete you can live out the rest of your days with the true isotypic decomposition in your pocket, ready to be deployed anytime.

Published by Jonathan Novak

Mathematician at UC San Diego.

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