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

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,

-
Okay. Thanks for that ;) – kηives May 22 '12 at 3:09
There is no representation on a link, there is a 2 by 2 matrix, an element of SU(2), on each link. The trace of the product is the matrix trace of the matrix product. Do you want to know how to simulate this? You need a metropolis step, and there are tricks for making good updates over big regions. – Ron Maimon May 22 '12 at 6:24

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

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 $SU(2)$.