# Which order multiplet of a given $SU(N)$ is real or complex?

I am studying the $$SU(2)$$ symmetric Lagrangian in particle physics.

$${\mathcal{L}} = (\partial_\mu \Phi)^\dagger (\partial^\mu \Phi) - (\mu^2 \Phi^ \dagger \Phi + \lambda (\Phi^ \dagger \Phi)^2).$$

In this the $$SU(2)$$ triplet taken consists of 3 real fields.

$$\Phi = \begin{bmatrix} \phi_1 \\ \phi_2 \\ \phi_3 \end{bmatrix}.$$

I am bit confused about why this triplet needs to be real? Can't it be also complex? Instead of a triplet can we take a doublet? If so, will it be real or complex?

In general if we are given a $$SU(N)$$ symmetric Lagrangian, is there any convention to choose real multiplet and which order multiplet should we choose so that it is a real multiplet?

• The fundamental/defining/spinor representation $${\bf 2}$$ of $$SU(2)$$ is a quaternionic/pseudoreal representation.
• More generally the $${\bf 2j\!+\!1}$$ irreducible representation of $$SU(2)$$ is (pseudo)real if the spin $$j$$ is a (half)integer, respectively.
• The fundamental/defining representation $${\bf N}$$ of $$SU(N)$$ for $$N>2$$ is a complex representation.