I have the problem of determining the most general form of a rank 2 tensor $t_{ij}$ (in 3 dimensions) satisfying:
- $t_{ij}$ is invariant under any rotation about the $z$-axis
- $t_{ij}$ is invariant under a rotation by $\pi$ about any axis in the $(x,y)$-plane through the origin
The question asks me to show the most general form of the above tensor is
$$t_{ij} = \alpha \delta_{ij} + \beta \delta_{i3}\delta_{j3}$$
where $\alpha$, $\beta \in \mathbb{R}$, and $\delta_{ij}$ is the Kronecker delta.
I think I have a fundamental misunderstanding about how to solve problems like this, since, taking the first bullet point, we would want
$$t_{ij} = \begin{pmatrix} \cos \theta &- \sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0 & 0 & 1 \end{pmatrix}t_{ij} \quad \forall \theta \in \mathbb{R}$$
So in particular, taking $\theta = \pi$ we would want
$$t_{ij} = \begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1 \end{pmatrix}t_{ij} \quad \forall \theta \in \mathbb{R}$$
which would make the entries of $t$ for $i=1,2$ to be $0$, and thus requiring $\alpha = 0$ in the given most general form, so this shouldn't be right.
I'm also struggling to see how I could check that the second condition is obeyed.
What am I doing wrong here?