The beautiful physical interpretation of the decomposition of the elasticity stress tensor into its irreducible representations (mentioned in context below) has inspired the following question on representation theory.
If you interpret a $3×3$ matrix $A$ as a direct product of vectors, $A = A_jB_k$, then breaking a matrix up into a symmetric and anti-symmetric matrix $A = A_s + A_a$ is a decomposition of a 9 component tensor into a sum of a smaller 6 component and 3 component tensor to get $\bf{3\otimes 3 = 6 \oplus 3}$. Apparently I just showed the matrix representation of orthogonal group $O(3)$ is reducible.
This is as far as you can go in $O(3)$, the group of rotations in 3-space, but if you have a unit determinant you can go further and use the trace to remove some redundancy. When we can do this we call it $SO(3)$, and can further reduce the symmetric matrix into a traceless 5-component symmetric matrix and a scalar multiple of the identity (by the trace) to get a $\bf{3\otimes3 = 5\oplus 3 \oplus 1}$, i.e. $A = A_{ts} + A_t + A_a$.
I apparently just described the tensor decomposition of the $SO(3)$ orthogonal group representation into its irreducible tensor representations just there, but any orthogonal group also has a spinor representation too.
This can be given a geometric interpretation: we decomposed a rank 2 tensor into the sum of a shearing, (a symmetric rank two tensor representing a unit ellipsoid), a rank 1 tensor rotation (a vector representing a rotation) and a rank $0$ tensor (a scaling factor). (pdf page 10)
1) Is there a similar easy decomposition of a 3×3 matrix into its spinor representations?*
I mean, starting with an explicit matrix can we do some easy process to it, like taking symmetric and anti-symmetric combinations as above, and magically end up with spinor representations and show how obvious their existence is? Just so that it's not so abstract.
2) How do we give a nice geometric interpretation to the decomposition of $SO(4)$?
In other words, how do we decompose matrices in $SO(4)$ in such a way that gives a geometric interpretation to every term, illustrates why the $SO(4) \cong [SU(2) \times SU(2)]/\mathbb{Z}_2$ should be obvious, and hopefully makes the symmetries of the Riemann curvature tensor look completely obvious?
(I'm really just hoping to derive and compare $SO(3)$ spinors with $SO(4)$ spinors, since they look so different in QM vs QFT, hence two parts to this question.)
Best Answer
When we say that $3\otimes 3 \approx 5 \oplus 3 \oplus 1$, this means that under an $SO(3)$ rotation, the first five components mix together alone "traceless symmetric tensor", so do the next 3, while the last component "the trace" is invariant.
If you prove that each of these three parts of the sum is indeed "irreducible", this means that this decomposition is UNIQUE! because in any other decomposition you are going to mix some of the "5" components with some of the "3", which would not work because these parts that you took out will definitely mix with the ones you left out (I'm trying to be as colloquial as possible).
We can also give physical arguments why a composite particle of two sub particles each having spin one (boson) cannot possibly be a half integer particle (fermion), which is what would happen if any of the representations being summed on the right hand side was even.
For the second part of your question then the unique decomposition of a tensor in $4D$ is
$4\otimes 4 \approx 9 \oplus \bar{3} \oplus 3 \oplus 1$
which are the same as before, but now the antisymmetric component splits into a dual and anti-self dual irreducible representations. (which from the point of view of the $SO(3)$ subgroup behave like a pseduo-vector and a polar vector)
REPLY TO YOUR COMMENT: Here is where it might get confusing.. If the group itself is a product, then its representation will also be $\otimes$ but you can't convert them into sums because the final answer should always be a product. This is why you see physicists denote a representation of $SU(2)\otimes SU(2)$ by $(n,m)$.. because it ensures that whatever manipulations you do, you will end up with something of the form $(n,m)$ even though this is just $n\otimes m$ in a sense.. Now to give you some examples of how representations of $SO(4)$ are related to those of $SU(2)\otimes SU(2)/Z2$: $4 \longrightarrow (2,2)$ for a four vector which can be thought of as a product representation of two spinors, each one in different part of the $SU(2)$. An antisymmetric 4-tensor: $(4\otimes 4)_A \longrightarrow (3,1)\oplus (1,3)$ and so on