Suppose $S^x$,$S^y$ and $S^z$ are the irreducible representations of $SU(2)$ in a complex vector space, V, of dimension N. By Burnside's theorem, $I$, $S^x$,$S^y$ and $S^z$ for a basis in the ring of endomorphims of V.
Now take $T_k$ as the generators of $SU(N)$. By the definition of $SU(N)$, $T_k$ is a subset of endomorphims in V.
Thus, I expect I should be able to write:
$ T_k=f_k(S^x,S^y,S^z,I) $
with $f_k$ being a polynomial.
Is this expecation correct. Did I understand Burnside's theorem and the definition of SU(N) correctly?
- If so, what can I say about $f_k$?
- What is the highest degree polynomial required?
- Do I need products like $S^xS^y$ or do monomials in $(S^i)^n$ suffice