Anti-triplet representation

Question

What is the anti-triplet representation of a group?

Specifically, what is the anti-triplet representation of $SU(3)$?

Answer 1

A mathematician will give more detailed and rigorous answer. From the physicist point of view, representation corresponds to the way the object transforms under a symmetry group $G$. Let us assign upper index to the object $a^{i}$, that transforms like a column under $G$: $$ a^{i} \rightarrow U_j^{i} \ a^{j} $$ This we will call the fundamental representation. For the case of $SU(N)$ the fundamental representation is a column of $N$-elements. The object with the lower index $b_i$ we will regard as transforming in antifundamental representation (it will reside in the dual vector space to the fundamental). $$ b_i \rightarrow (U^*)_{i}^{ j} b_j $$ This object will transform as a row, or ,looking in another way, as a column, but under the hermitian conjugated matrix $U^{\dagger}$. For the case of $SU(N)$ think about it as row of $N$ complex numbers.

Answer 2

The anti-triplet representation $\bar{\bf 3}$ is presumably the complex conjugate representation of the defining/fundamental $SU(3)$ representation ${\bf 3}$.