Difference between dual representation and antifundamental representation? And why is it interesting?

I'm confused about antifundamental and dual representations.

Let $$J_a$$ be a set of generators, which satisfy $$[J_a, J_b] = i f_{abc} J_c$$, so that $$e^{i\alpha J} \in \mathcal{SU}(n)$$.

For the complex conjugate we find: $$[J_a^*, J_b^*] = -i f_{abc}J_c^*$$, so if we let $$\overline{J}=-J^*$$ we find that $$[\overline{J}_a, \overline{J}_b] = i f_{abc}\overline{J}_c$$. For the hermitian operators $$J_a$$ with real Eigenvalues $$j_a$$, we then find that the Eigenvalue changes sign. This is the same behaviour as is we were to replace the observed particle with it's anti particle, where all quantum number are multiplied by $$(-1)$$. $$J*$$ is known as the antifundamental representation and used to describe anti particles. We can use this to show, that there exists an similarity transformation in $$\mathcal{SU}(2)$$, so that after reordering the nucleon doublet, we can use the pauli matrices as well for the anti-nucleon doublet.

However, in $$\mathcal{SU}(3)$$ I've read, that they use the dual representation of the Gell-Mann matrices $$\overline{\lambda}_a = -\lambda_a^\mathrm{T}$$, which is the sign inverted transpose.

1. What is the difference between the antifundamental representation and the dual representation?
2. Why is the dual space of interest in this matter?
3. Which should I use to describe anti particles? And why?
• Important note: Mathematician and Physicists definitions of the exponential map differ by a factor of $i$, and hence the mathematicians generators of $\mathcal{SU}(N)$ are skew-hermitian, not hermitian, since another $i$ is needed on the generators to have overall no change. This causes even more confusion because dual is then related to the complex conjugate differently in the two situations. For the mathematician's skew-hermitian generators is it obvious that dualizing ($\lambda \rightarrow -\lambda^T$) is the same as complex conjugation. Commented Apr 13, 2023 at 21:39
• In nuce: stick with the definition of the dual (also known as contragredient) representation being given by $-\lambda^T$ (since it is immune to multiplication by $i$) and you should be fine for all semi-simple Lie groups. As doetoe says, for compact Lie groups, dualizing is the same as complex conjugating (more discussion here), so it is common to define the dual using conjugation instead. For more general groups it is best to keep the notions of dualizing and complex conjugating separate, as they in general give different representations. Commented Apr 13, 2023 at 21:51

The antifundamental representation is just the complex conjugate representation of the fundamental representation. In the case of unitary representations the complex conjugate representation is isomorphic to the dual representation. Because of this, in the context of quantum field theory these terms are often used interchangeably.

Finally, in the special case of $$\mathrm{SU}(2)$$ (unlike the case of $$\mathrm{SU}(N)$$ for $$N > 2$$) the fundamental representation and the antifundamental representation are isomorphic as well.