The basic question I guess can be formulated as - given two integers $N_f$ and $N_c$ what are the ways in which the fundamental and the anti-fundamental representations of $U(N_f)$ be combined to get 1-dimensional representations of either $U(N_c)$ or $SU(N_c)$.

Apparently that is what goes into the following classification of particle physics as,

If you have $N_f$ fields in the fundamental representation of $U(N_f)$ then apparently these can't be combined (tensored?) into an $U(N_c)$ invariant (gauge singlet).

But the same $N_f$ fields can be combined into "baryons" - gauge singlets of $SU(N_c)$ as, $\epsilon_{i_1\dots i_{N_f-N_c}j_1\dots j_{N_c}}\epsilon^{a_1\dots a_{N_c}}$ $\prod_{k=1}^{N_c} \phi^{j_k}_{a_k}$

If with the same $SU(N_c)$ the $N_f$ fields happen to be in the adjoint of $U(N_f)$ then there exists forms invariant under $SU(N_c)$ given as $Tr[\prod_{k=1}^n \phi_{i_k}]$ (for any $n$ of these $N_f$ fields)

If one has a pair of fields in the fundamental and the anti-fundamental of $U(N_f)$ then the gauge invariant operators under $U(N_c)$ are given as the "mesons" - $\phi^i_a \bar{\phi}^a_j$ (where $a$ is the $N_c$ index and $i,j$ is the $N_f$ index)

I guess there is an uniqueness about the gauge invariant objects created for each flavour combination given. I guess this "fundamental theorem of invariant theory" in some sense guarantees this uniqueness.

It would be great if someone can explain this.

(.googling "fundamental theorem of invariant theory" didn't yield any clear answer..)

samegroup $G$. – Victor Protsak May 14 '13 at 5:46