One-Plaquette Action and SU(2)'s Irreducible Representations

Question

I have a typical single-plaquette partition function for a gauge-field $$ Z=\int [d U_{\text{link}}] \exp[-\sum_{p} S_{p}(U,a)]$$ with $U$ as the product of the the $U$'s assigned to each link around a plaquette. Now the $U$'s are irreducible representations of my group elements, which in my case is SU(2), and lets take the 1/2 representation as an example, then define the character as $\Xi_{r}\equiv \text{Tr}[U]$ which for our case is $\Xi_{1/2}=\text{Tr}[U]$. Now I have to take the product of these representations, however (and here's my question), how do I know which group elements/that-element's-representation to assign to each link?

I'm not sure how to compute the Trace without knowing first how to do the product of the link's representations, but I don't even know how to assign the elements to the links.

Thanks,

Answer 1

I'm not completely sure what OP is asking(v1). However here is my interpretation.

OP asks:

How do I know which group elements [...] to assign to each link?

The group element $U_{\ell}\in SU(2)$ affiliated with a link $\ell\in L$ is not fixed. One is supposed to integrate over all possible group values of $U_{\ell}\in SU(2)$. Phrased differently, the link variables $(U_{\ell})_{\ell\in L}$ are the dynamical variables of the model.

The integration measure in the integral for $Z$ reads

$$ [dU]~=~ \prod_{\ell\in L} dU_{\ell}$$

Here $dU_{\ell}$ typically denotes the Haar measure for $SU(2)$.