  Hodge representations were introduced by Green-Griffiths-Kerr to classify the
Hodge groups of polarized Hodge structures, and the corresponding Mumford-Tate
subdomains of a period domain. The purpose of this article is to provide an
exposition of how, given a fixed period domain $\mathcal{D}$, to enumerate the
Hodge representations corresponding to Mumford-Tate subdomains $D \subset
\mathcal{D}$. After reviewing the well-known classical cases that $\mathcal{D}$
is Hermitian symmetric (weight $n=1$, and weight $n=2$ with $p_g = h^{2,0}=1$),
we illustrate this in the case that $\mathcal{D}$ is the period domain
parameterizing polarized Hodge structures of (effective) weight two Hodge
structures with first Hodge number $p_g = h^{2,0} = 2$. We also classify the
Hodge representations of Calabi-Yau type, and enumerate the horizontal
representations of CY 3-fold type. (The "horizontal" representations those with
the property that corresponding domain $D \subset \mathcal{D}$ satisfies the
infinitesimal period relation, a.k.a. Griffiths' transversality, and is
therefore Hermitian.)
