Rotation invariant bilinear form There is a passage in a book I am reading that I don't understand, regarding invariance under group transformations.
The author starts by saying the functions $\{\phi_\alpha\}$, with $\alpha\in\{-1,0,1\}$, span the triplet representation of $SO(3)$. This means that under rotations they transform as $\phi'_\alpha=\sum_\beta D_{\alpha\beta}\phi_\beta$, where $D$ are usual rotation matrices.
This I think I understand and the picture in my mind is just three ortogonal vectors (is this appropriate?)
Then the author says, without further explanations, that the bilinear form $\sum_\alpha \phi_\alpha\phi_{-\alpha}(-1)^{\alpha-1}$ is invariant under rotations.
Now I am completely lost, because I thought the invariant bilinear form should be the usual dot product, $\sum_\alpha \phi_\alpha^2$.
Can someone clarify this, please?

If the functions $\phi_\alpha$ span an irreducible representation, they must be linear combinations of spherical harmonics $Y_{l,m}$. Let us assume they are exactly the spherical harmonics, for simplicity. For the triplet index $\alpha$ is $m$ and $l=1$.
What your book is telling you is that $$\sum_{m=-1}^1 Y_{1,m}Y_{1,-m}(-1)^m$$ should be rotation invariant, i.e. a scalar.
This is true and it boils down to two well known properties of spherical harmonics. First, $Y_{l,-m}=(-1)^mY_{l,m}^*$, so that your expression becomes $\sum_{m=-1}^1 Y_{1,m}Y_{1,m}^*$. Second, the addition theorem,
$$ \sum_{m=-l}^l Y_{l,m}(\vec x)Y_{l,m}^*(\vec y)=\frac{2l+1}{4\pi}P_l(\vec x \cdot \vec y),$$ where $P_l$ are Legendre polynomials.
