Unitary Group Representations

Unitary Group Representations

Definition: Let $G$ be a group and let $(V, \rho)$ be a group representation of $G$ where $V$ is an inner product space with inner product $\langle - , - \rangle$. We say that $(V, \rho)$ is a Unitary Representation of $G$ if for all $g \in G$ and for all $u, v \in V$ we have that $\langle [\rho(g)](u), [\rho(g)](v) \rangle = \langle u, v \rangle$.
Theorem 1: Let $G$ be a group. If $(V, \rho)$ is a unitary representation of $G$ then this representation is either irreducible or decomposable.
  • Proof: Suppose that $(V, \rho)$ is not irreducible. We aim to show that then $(V, \rho)$ is decomposable.
  • Since $(V, \rho)$ is reducible there exists a nonzero, proper subspace $W \subset V$ that is $G$-invariant. Since $V$ is an inner product space, and $W$ is a subspace of $V$ we can consider the set:
\begin{align} W^{\perp} = \{ v \in V : \langle v, w \rangle = 0, \: \forall w \in W \} \end{align}
  • We know from linear algebra that $W^{\perp}$ is a subspace of $V$ and that $V = W \oplus W^{\perp}$. Now observe that for all $w \in W$:
\begin{align} \langle [\rho(g)](v), w \rangle &= \langle v, [\rho(g)]^*(w) \rangle \\ &= \langle v, [\rho(g)]^{-1}(w) \rangle \\ &= \langle v, [\rho(g^{-1})](w) \rangle \\ &= 0 \end{align}
  • (Where the last equality comes from the fact that $\rho(g^{-1})(w) \in W$ since $W$ is $G$-invariant. Thus, $W^{\perp}$ is a subrepresentation of $(V, \rho)$. So $(V, \rho)$ is decomposable. $\blacksquare$

As a corollary to the above theorem, we see that irreducibility and indecomposability are equivalent for unitary group representations of a group $G$.

Corollary 2: Let $G$ be a group. If $(V, \rho)$ is a unitary group representation of $G$ then $(V, \rho)$ is irreducible if and only if it is indecomposable.

For UNITARY group representations:

\begin{align} \mathrm{Irreducible} \: \Leftrightarrow \: \mathrm{Indecomposable} \end{align}
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