What is meant by "${\bf N}$ of a group"?
The ${\bf N}$ of a group is in fact a shorthand for the $N$-dimensional irreducible representation of this group.
Is is just shorthand for an ${\bf N}$ representation? If so, what
exactly is an ${\bf N}$ representation of a given group?
The group elements are abstract operations defined by how they act on given objects. For example, the rotation group in three dimensions, $\mathrm{SO(3)}$, is formed by elements that rotate coordinate systems in such a way that the length of any vector is invariant. In order to make things more explicit we assign linear representations to these groups, i.e. we map the group elements into matrices acting upon some vector space $\mathbb V$. If $\mathbb V$ is $N$-dimensional so it is the group representation.
If all $N$-dimensional matrices representing the group elements can - by a similarity transformation - be written in a block diagonal form then the representation is said to be reducible. Otherwise it is called irreducible (or simply an irrep.) and can be labelled by $\bf N$ which denotes its dimension. For example, general rotation in the plane, which constitute the group $\mathrm{SO(2)}$, can be written simply as $e^{i\theta}$ giving a one-dimensional irreducible representation. On the other hand, the two-dimensional representation
$$
\begin{bmatrix}
\cos\theta&-\sin\theta\\
\sin\theta&\cos\theta\\
\end{bmatrix}.
$$
is reducible since it can be put in the diagonal form
$$
\begin{bmatrix}
e^{i\theta}&0\\
0&e^{-i\theta}\\
\end{bmatrix},
$$
by making use of a similarity transformation generated by
$$
\frac{1}{\sqrt 2}
\begin{bmatrix}
1&1\\
-i&i\\
\end{bmatrix}.
$$
We should label that one-dimensional irreducible representation this by $\bf 1$ and therefore the two-dimensional reducible representation is labeled by $\bf 1\oplus \bf 1$, where the $\bf 1$ refers to each of the one dimensional block that can be written after a similarity transformation. This representation is actually acting upon a direct sum of two vector spaces of dimensions $1$.
- How can I work out / write down such a representation concretely, like the Pauli matrices for $SU(2)$? I'd be grateful for a simple example.
- What does it mean when something "transforms like the ${\bf N}$"?
To work out the irreducible representations we need to deal with the algebra instead of the group. Among all elements of a Lie group there are special ones that can be used to generate any other. They are called generators of the group and they satisfy a particular structure, called Lie algebra. For example, the group $\mathrm{SU}(2)$ has a Lie algebra $\mathfrak{su}(2)$ whose generators are $T_a$, $a=1,2,3$, satisfying
$$[T_a,T_b]=i\epsilon_{abc}T_c.$$
A representation $R$ of these abstract elements has to preserve this structure, i.e.,
$$[R(T_a),R(T_b)]=i\epsilon_{abc}R(T_c),$$
where $R(T)$ shall be understood as an $N$-dimensional matrix.
From the Lie algebra one can obtain all possible representations. This is usually done by writing the generators in the so-called Cartan-Weyl basis which decomposes the algebra into the Cartan subalgebra (the maximal set of self-commuting or diagonalizable generators) and the step or ladder operatores. The states of a given irreducible representation are then given by the eigenvectors of the Cartan generators. Clearly these states are $N$-dimensional vectors given that the algebra representation is $N$-dimensional. So when we say that something - a field for instance - transforms like the $\bf N$ of an algebra we mean that this object is mapped to a column matrix with $N$ entries whose basis is given by the eigenvectors mentioned above. For example, the $\mathfrak{su}(2)$ algebra has only one step operator, $T_3$. For a two dimensional irrep. the matrix $R(T_3)$ has two eigenvectors. A field transforming like $\mathbf N$ - or simply as a doublet - is $\phi$, such that
$$R\phi=
\begin{bmatrix}
a&b\\
c&d\\
\end{bmatrix}
\begin{bmatrix}
\phi_1\\
\phi_2
\end{bmatrix}
=\begin{bmatrix}
\phi_1'\\
\phi_2'
\end{bmatrix}.
$$
One can show for instance that the $\mathfrak{su}(2)$ algebra has $N$-dimensional representations for any integer $N$. The classical algebras $\mathfrak{su}(n)$, $\mathfrak{so}(n)$ and $\mathfrak{sp}(n)$ have at least the singlet, the defining and the adjoint representations. The singlet is the one dimensional representation, i.e. they are only numbers. Notice that the only possibility numbers can satisfy a non-trivial algebra is that they are all zero. They are useful in physics when something does not transform at all. The defining representation is the $n$-dimensional, e.g. the three-dimensional representation for a quark transforming under the flavor $\mathfrak{su}(3)$. When the vector field $\mathbb V$ is the algebra itself the representation is called adjoint. In this case, the dimension of the algebra equals the dimension of the representation. The gauge fields transform under this representation of the gauge groups. For example, the algebra $\mathfrak{su}(5)$ has $24$ generators so the $\bf{24}$ is the adjoint representation of $\mathfrak{su}(5)$.
Once we know the representation $R(T)$ for a Lie algebra we can induce it to the group by means of an exponential operation,
$$R(g)=\exp\left[i\phi R(T)\right],$$
where $g$ denotes the group element. Notice that if we have a singlet of the algebra, the singlet of the group turns out to be just the number $1$.
There is although some subtleties when going from the algebra to the group. Starting from a given Lie algebra and assigning a given representation one can ends up with different Lie groups. So for adjoint representation of $\mathfrak{su}(2)$ the group generated turns out to be $\mathrm{SO}(3)$ instead of $\mathrm{SU}(2)$.
