Simplification of $\sum_{a=1}^3\sigma^a_{\alpha\beta} S^a_{\gamma\delta}$ where $\sigma^a,S^a$ are representations of $SU(2)$?

Question

As the question asks, I am dealing with a problem where I'd like to simplify

$$\sum_{a=1}^3 \sigma^a_{\alpha\beta} S^a_{mn}$$ where $\sigma^a$ are the spin-1/2 Paulis and $S^a$ are some higher-spin representation of $SU(2)$. For the case where $S^a = \sigma^a$, we know $$\sum_{a=1}^3 \sigma^a_{\alpha\beta} \sigma^a_{\gamma\delta}=2\delta_{\alpha\delta}\delta_{\beta\gamma} - \delta_{\alpha\beta}\delta_{\gamma\delta}$$

and I'm searching for a higher-spin analog of this formula. There is probably some representation-theoretic solution.

Answer 1

Since your title asks for a mere "simplification", and not a Fierz-transposition, argued to be extraordinarily unlikely and problematic in the comments, I'll repeat the undergraduate QM procedure for your cross expression, as it summarizes the "easy" part of "adding" spins.

You have a spin doublet (2 states operated upon by the generators $\vec \sigma/2$) tensored to a multiplet of 2s+1 states acted upon by $\vec S$. So the tensor product space everything fits in is 4s+2 -dimensional. You already know this representation of SU(2) reduces to a direct sum of a 2s+2 -plet acted upon by spin s+1/2 generators, with a 2s-plet acted upon by spin s-1/2 operators. In your prototype example, you have s=1/2.

Your coproduct (composite reducible rep operators) then squares to just $$ (\vec {\sigma \over 2}\otimes {\mathbb I}_{2s+1} +{\mathbb I}_{2} \otimes \vec S )^2 =\frac{3}{4} {\mathbb I}_2 \otimes {\mathbb I}_{2s+1} +s(s+1) {\mathbb I}_2 \otimes {\mathbb I}_{2s+1} + \vec \sigma \cdot \otimes \vec S, $$ where I have inserted the values of the quadratic Casimirs of each component representation, and the last, non diagonal term is what you are interested in.

But you also know this representation reduces through a Clebsch matrix similarity transformation to two separate blocks of the spin s+1/2 and the spin s-1/2 representations, so the above Casimir reduces to the following upon this (undisclosed!!) similarity transformation, $$ (s^2+2s+3/4) {\mathbb I}_{2s+2} \oplus (s^2-1/4) {\mathbb I}_{2s} . $$

So, regardless of the basis involved, these two expressions have an evident action when acting on the big, spin s+1/2 multiplet, and small, spin s-1/2 multiplet, respectively.

On the big multiplet, the eigenvalue of $\vec \sigma \cdot \vec S$ is $$ -3/4- s(s+1)+ (s^2+2s+3/4)=s, $$ whilst on all states of the the small multiplet, the eigenvalue is $$ -3/4- s(s+1)+ (s^2 -1/4)=-s-1 . $$

Note for your conventional paradigm, s=1/2, the corresponding eigenvalues are the celebrated 1/2 and -3/2, respectively.