Nonlinear representation of SU(2) and SU(4) Consider a nonlinear representation of a group acting on the space of real vector $\phi_i$ in the form:
\begin{equation}
\phi_i\rightarrow \sum_jM_{ij}\phi_j+\delta\phi_i
\end{equation}
where $M$ is a matrix whose entries are all integers and has determinant $\pm 1$, and the real vector $\delta\phi_i$ is not a zero vector in general. Can anyone tell me what this kind of representations of $SU(2)$ and $SU(4)$ are or give me relevant reference?
 A: This is called an affine representation.  In general, any affine transformation of the form
$$
\textbf{v} \;\mapsto\; M\textbf{v} + \textbf{b}
$$
where $M$ is a matrix and $\textbf{b}$ is a vector can be represented as acting on $n+1$-dimensional vectors $(v_1,\ldots,v_n,1)$ via the block matrix
$$
\begin{bmatrix}M & \textbf{b} \\ \textbf{0}^T & 1\end{bmatrix}.
$$
Note that $\textbf{e}_{n+1}$ is an eigenvector of the transpose of this matrix, with eigenvalue $1$.
If $G$ is a compact Lie group, it turns out that any affine representation of $G$ fixes some point $\textbf{a}$.  It follows that, for any such representation of $G$, each element $g\in G$ acts as an affine transformation of the form
$$
\textbf{v} \;\mapsto\; \rho(g) (\textbf{v}-\textbf{a}) + \textbf{a}
$$
where $\rho\colon G\to GL(n)$ is some linear representation of $G$.  (So the affine representation is essentially just a linear representation centered at $\textbf{a}$.) This follows from the matrix representation of affine transformations given above.  In particular, in this matrix representation, the subspace of vectors whose last coordinate is zero is an invariant linear subspace.  Since $G$ is compact, this must have a complementary one-dimensional invariant subspace, which contains a vector of the form $(a_1,\ldots,a_n,1)$.  It is easy to see this vector must be fixed, and hence every affine representation of a compact Lie group fixes a point.
Incidentally, a different, more geometric proof that there must be a fixed point is to simply use the Haar measure to find the center of mass of the orbit of any point, i.e.
$$
\textbf{a} \;=\; \frac{1}{\mu(G)} \int_G g(\textbf{v})\,d\mu(g)
$$
where $\textbf{v}$ is any vector in $\mathbb{R}^n$ and $\mu$ is the Haar measure on $G$.
