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edit approved Feb 2, 2021 at 10:12
gmvh
edit approved Feb 2, 2021 at 10:12
gmvh
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Dimension of intertwiner space: finite dimensinoal-dimensional representations of compact groups

Let $G$ be a compact group with finite dimensional-dimensional, real representations $\phi$ and $\psi$ on $V$ and $W$ respectively. (e.g. $V = \mathbb{R}^m$, $W = \mathbb{R}^n$.) Is it true that, as is the case for finite groups, the dimension of the intertwiner space of the two representations is equal to the inner product of the characters? That is, do we still have

$$ \dim \text{Hom}_G(V, W) = \int_G \text{d}\lambda(g) \text{Tr}(\phi(g))\text{Tr}(\psi(g)) $$ where $\lambda$ is the Haar measure on $G$?

As an aside, do the proposed representations always exists? Do they exist if I insist they are unitary/orthogonal?

Dimension of intertwiner space: finite dimensinoal representations of compact groups

Let $G$ be a compact group with finite dimensional, real representations $\phi$ and $\psi$ on $V$ and $W$ respectively. (e.g. $V = \mathbb{R}^m$, $W = \mathbb{R}^n$.) Is it true that, as is the case for finite groups, the dimension of the intertwiner space of the two representations is equal to the inner product of the characters? That is, do we still have

$$ \dim \text{Hom}_G(V, W) = \int_G \text{d}\lambda(g) \text{Tr}(\phi(g))\text{Tr}(\psi(g)) $$ where $\lambda$ is the Haar measure on $G$?

As an aside, do the proposed representations always exists? Do they exist if I insist they are unitary/orthogonal?

Dimension of intertwiner space: finite-dimensional representations of compact groups

Let $G$ be a compact group with finite-dimensional, real representations $\phi$ and $\psi$ on $V$ and $W$ respectively. (e.g. $V = \mathbb{R}^m$, $W = \mathbb{R}^n$.) Is it true that, as is the case for finite groups, the dimension of the intertwiner space of the two representations is equal to the inner product of the characters? That is, do we still have

$$ \dim \text{Hom}_G(V, W) = \int_G \text{d}\lambda(g) \text{Tr}(\phi(g))\text{Tr}(\psi(g)) $$ where $\lambda$ is the Haar measure on $G$?

As an aside, do the proposed representations always exists? Do they exist if I insist they are unitary/orthogonal?

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user27182
user27182
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Dimension of intertwiner space: finite dimensinoal representations of compact groups

Let $G$ be a compact group with finite dimensional, real representations $\phi$ and $\psi$ on $V$ and $W$ respectively. (e.g. $V = \mathbb{R}^m$, $W = \mathbb{R}^n$.) Is it true that, as is the case for finite groups, the dimension of the intertwiner space of the two representations is equal to the inner product of the characters? That is, do we still have

$$ \dim \text{Hom}_G(V, W) = \int_G \text{d}\lambda(g) \text{Tr}(\phi(g))\text{Tr}(\psi(g)) $$ where $\lambda$ is the Haar measure on $G$?

As an aside, do the proposed representations always exists? Do they exist if I insist they are unitary/orthogonal?