I need to perform various Tensor contractions of SU(N) groups with the Levi-Civita tensor. In particular I need to explicitly check if certain contractions cancel out or not(the actual answer of the contraction is irrelevant). I am coding this process on the computer, and thus need an explicit way of doing these contractions. That is, the SU(N) reps need to be represented in array form with all relevant symmetries.
For example, suppose $u$ is the antifundamental ($\bar{3}$) of $SU(3)$ with young diagram:
and $p$ is a $6$ of the $SU(3)$ group with young diagram
As a tensor they are $u^{ab}$ which is antisymmetric in it's indices and $p^{cd}$ which is completely symmetric in its indices.
My question is: how could I represent $u^{ab}$ as a matrix as well as $p^{cd}$?
Furthermore how can I represent the adjoint ($8$) of $SU(3)$ which now has mixed symmetry:
$q^{[eg]f}$.
Ultimately I want to be able to do this for any rep of any SU(N) group. This way, it makes coding the contractions explicitly easier.
An example of such a contraction would be say $\bar{3}\otimes\bar{3}\otimes\bar{3}$. i.e :
$$ u\otimes u\otimes u $$ which will give you the following decomposition
$1 + 2(8) + \bar{10}$.
For my purposes, I want to contract the singlet with the Levi-Civita symbol, thus giving:
$$ u^{ab}u^{cd}u^{ef} \epsilon_{abd}\epsilon_{cef}. $$ The above should cancel as all the $u's$ are the same.
I want to show this computationally, however I need to imput a specefic form of the matrix of $u$ to perform such a sum. In physics, reps of of various $SU(N)$ groups are used to encode particles. Taking the product of various reps and contracting the singlet form of the product with the Levi-Civita tensor is of some interest, and is what I am working on. Hence the above question.
There are various methods for constructing $su(3)$ basis states. Possibly the least painful one from an operational perspective was given by Rowe, D. J., B. C. Sanders, and H. de Guise. "Representations of the Weyl group and Wigner functions for SU (3)." Journal of Mathematical Physics 40.7 (1999): 3604-3615 and relies on duality between $SU(3)$ and $SU(2)$ irreps in symmetric irreps of $SU(6)$. If $$ \vert s_im_i\rangle = \frac{(a_{i1}^\dagger)^{s_i+m_i}(a_{i2}^\dagger)^{s_i+m_i}}{\sqrt{(s_i+m_i)!(s_i-m_i)!}}\vert 0\rangle $$ is an $su(2)$ state written in terms of creation operators acting on the vacuum, then the linear combination \begin{align} &\vert(\lambda,\mu)\nu_1\nu_2\nu_3:I\rangle\nonumber \\ &=\sum_{m_1m_2m_3}C^{I,N}_{\frac{1}{2}\nu_3m_3;\frac{1}{2}\nu_2m_2} C^{\frac{1}{2}\lambda,\frac{1}{2}\lambda}_{,N;\frac{1}{2}\nu_1m_1} \vert \textstyle\frac{1}{2}\nu_1m_1\rangle \vert \textstyle\frac{1}{2}\nu_2m_2\rangle\vert \textstyle\frac{1}{2}\nu_3m_3\rangle \end{align} where $C^{J,M}_{j_1m_1;j_2m_2}$ is an $su(2)$ Clebsch-Gordan coefficient, and $(\lambda,\mu)$ are the Dynkin labels. Thus: $$ (\bar 3)\leftrightarrow (0,1)\, ,\qquad (6)\leftrightarrow (2,0)\, ,\qquad (8)\leftrightarrow (1,1) $$ and I think $(\overline{10})$ would be $(0,3)$.
In general constructing states is quite messy. In the case of the fundamental it's not that hard, and in case of $(\bar 3)$ it's also not so bad since it's just a determinant, but there is no easy way to get the other ones.
It is also possible to use Clebsch-Gordan technology and at least one online program for the $su(n)$ coupling.