# What is the “fundamental theorem of invariant theory” ?

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..)

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Your question doesn't contain enough information for a sensible answer until you also specify the $SU(N_c)$ representation of the fields and also their statistics (bosons or fermions). – Jeff Harvey May 13 '13 at 22:48
@Jeff Harvey Thanks for your comments. The question arises from trying to understand the classification given on the top of page 10 of arXiv:0704.3740 There it doesn't seem that they have specified the gauge representation of the matter fields. (..also I wonder if this uniqueness of the gauge singlet combinations has anything to do with the fact that there they also want the states to be BPS/superconformal primary...) I didn't understand why they need the chiral primary to be a homogeneous polynomial in such gauge singlets. It would be great if you can fill in what is being kept implicit there – user6818 May 13 '13 at 23:08
This question reminds me of the Jabberwocky poem :-) – Mariano Suárez-Alvarez May 13 '13 at 23:24
@user6818 As in the question is not well posed to start with and second it is not written in language that most mathematicians will understand. I partially understand what you are asking because I happen to be a physicist. I'd suggest that you either ask the question on physics stack exchange or make the effort to translate your question into a precise mathematical question framed in language that mathematicians will understand. Otherwise your question will be and should be closed since this is a site for research level math questions. – Jeff Harvey May 14 '13 at 0:05
I agree that the question resembles Jabberwocky. What is the relation between representations of $U(N_f)$ and $U(N_c)$? Standard linear algebra operations (direct sums, tensor products, symmetrizations, etc) would transform representations of a group $G$ (here $U(N_f)$) into representations of the same group $G$. – Victor Protsak May 14 '13 at 5:46