Math 202B: Lecture 24

*** Problems in this lecture Due March 15 ***

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Let $G$ be a finite group, and let $\Lambda$ be a set parameterizing irreducible unitary representations of $G$ up to isomorphism. We know that $\Lambda$ is a finite set whose cardinality is bounded by the number of conjugacy classes in $G.$ For each $\lambda \in \Lambda,$ let $(V^\lambda,\varphi^\lambda)$ be an irreducible unitary representation of $G$ , and denote by

$\chi^\lambda(g) = \mathrm{Tr}\ \varphi^\lambda(g), \quad g \in G,$

the corresponding character. Then, $\{\chi^\lambda \colon \lambda \in \Lambda\}$ form an orthogonal set of functions in the class algebra $\mathcal{K}(G)$ ,

$\langle \chi^\lambda,\chi^\mu\rangle= \delta_{\lambda\mu}|G|.$

We do not yet know that $\{\chi^\lambda \colon \lambda \in \Lambda\}$ spans $\mathcal{K}(G),$ but we will get there soon.

Today we will discuss the representation-theoretic analogue of prime factorization. In fact, decomposing representations is easier than decomposing numbers because our set of irreducible objects $\{V^\lambda \colon \lambda \in \Lambda\}$ is finite, and there is a formula for the multiplicity of a given irreducible in a given representation.

Let $(V,\varphi)$ be any finite-dimensional unitary representation of $G.$ Either this representation is irreducible, or not. If it is, then $(V,\varphi)$ is isomorphic to $(V^\lambda,\varphi^\lambda)$ for some $\lambda \in \Lambda).$ If not, then it contains a proper subrepresentation, i.e. a nonzero subspace $V_1<V$ which is $G$ -stable.

Theorem 24.1 (Maschke’s Theorem). The orthogonal complement $V_1^\perp$ is also $G$ -stable.

Proof: For any $v_0 \in V_1^\perp$ and $v_1 \in V_1$ we have

$\langle \varphi^\lambda(g)v_0,v_1\rangle = \langle v_0,\varphi^\lambda(g^{-1})v_1\rangle = 0$

for all $g \in G.$ $\square$

We now have a decomposition $V=V_1 \oplus V_2$ of $V$ into two proper subrepresentations. Repeat this process on each of the “factors” $V_1$ and $V_2$ , and after finitely many steps you will produce a binary tree whose leaves are irreducible subrepresentations of $V$ . Each of these leaves is necessarily isomorphic to one of the “primes” $(V^\lambda,\varphi^\lambda).$ We thus have a “prime factorization”

$V \simeq \bigoplus\limits_{\lambda \in \Lambda} \mathrm{mult}(V^\lambda,V)V^\lambda,$

where $\mathrm{mult}(V^\lambda,V)$ is the number of leaves in the tree isomorphic to $V^\lambda.$ This is called the isotypic decomposition of $V$ . It remains to verify that however we decompose $(V,\varphi)$ into irreducibles, the end result will be the same.

Problem 24.1. State and prove a uniqueness theorem for “the” isotypic decomposition of a representation (hint: character orthogonality).

Theorem 24.2. We have

$\mathrm{mult}(V^\lambda,V)=\frac{1}{|G|}\langle \chi^\lambda,\chi^V\rangle.$

Proof: Taking traces on each side of the isotypic decomposition yields

$\chi^V = \sum\limits_{\lambda \in \Lambda} \mathrm{mult}(V^\lambda,V)\chi^\lambda,$

now use character orthogonality. $\square$

Problem 24.2. Prove that $\mathrm{mult}(V^\lambda,V) = \dim\mathrm{Hom}_G(V^\lambda,V).$

Now we are in position to prove that the character of a representation really does characterize it, up to isomorphism.

Theorem 24.3. Two unitary representation $(V,\varphi)$ and $(W,\psi)$ of $G$ are isomorphic if and only if $\chi^V=\chi^W.$

Proof: It is clear that isomorphic representations have the same character. For the converse, the fact that $\chi^V=\chi^W$ implies that $\langle \chi^\lambda,\chi^V\rangle = \langle \chi^\lambda,\chi^W\rangle$ for all $\lambda \in \Lambda,$ so that $V$ and $W$ have the same isotypic decomposition. $\square$