Example of a group action G on a vector space V that fails to be linear (i.e. fails to be a linear representation).

Question

I have seen the following definition of a linear group representation from C. Lent's notes on Representation Theory:

A linear representation $ρ$ of $G$ on a complex vector space $V$ is a set-theoretic action on $V$ which preserves the linear structure, that is:

1. $ρ(g)(v_1 + v_2) = ρ(g)v_1 + ρ(g)v_2, ∀v_1,v_2 ∈ V$
2. $ρ(g)(kv) = k · ρ(g)v, ∀k ∈ C, v ∈ V$

This definition would imply that there exist actions of $G$ on a vector space $V$ which fail the preserve the linear structure of $V$ in this sense. Can anyone provide a good example?

Answer 1

After some more thinking I have thought of an example:

Consider the translation action of the integers $\mathbb{Z}$ as a group under additon on $\mathbb{R^2}$ given by: $n * (x,y) = (x+n, y)$.

We can verify that this does indeed form a group action:

1. $m * (n * (x,y)) = m * (x + n, y) = (x + n + m, y) = (m+n) * (x,y)$
2. $0 * (x,y) = (x + 0, y) = (x,y)$

However, this action does not respect scalar multiplication for $k \neq 0$ and $n \neq 0$ since this gives $n * (k(x,y)) = n * (kx, ky) = (kx + n, ky) \neq (kx + kn, ky) = k(x+n, y) = k (n * (x,y))$.

Answer 2

If I remember rightly, a standard sort of example would be: take the space of matrices $$ A(k)=\begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix}, $$ for real $k$. This is a group under multiplication with $A(k)A(-m) = A(k-m)$. Define an action of $\rho$ on $\mathbb{R}$ by $$ \rho(A(k))(v) = v+k $$ for any $v \in \mathbb{R}$. Now, this is an action since $$\rho(A(k))(\rho(A(m))(v)) = (v+m)+k = v+(k+m) = \rho(A(k)A(m))(v) \quad \text{and} \quad \rho(A(0))(v)=v,$$ but $$ \rho(A(k))(u+v) = u+v+k \neq u+k+v+k = \rho(A(k))(u)+\rho(A(k))(v), $$ and $$ \rho(A(k))(\lambda v) = \lambda v+k \neq \lambda(v+k) = \lambda \rho(A(k))(v), $$ so none of the linear structure remains.

Answer 3

An important example coming from representation theory is the so called "dot" action of a Weyl group for a semisimple Lie algebra acting on the dual Cartan. It is given by the formula $w \cdot \lambda = w(\lambda+\rho)-\rho$ where the non-dot action is a linear one, so it can be thought of as a shifted linear action where we have moved the origin to $-\rho$.

Answer 4

Here are some examples:

• The most extreme example is the symmetric group on the underlying set of $V$.
• The affine group of $V$, which is generated by invertible linear maps and translations. This can be generalized by replacing ${\rm GL}(V)$ with any matrix subgroup $G$ to get $V\rtimes G$. In particular $G=1$ gives us the group $V$ acting on itself by translations.
• Considering $\Bbb R^n$ as a real $n$-dimensional Riemannian manifold, consider
  • The isometry group of invertible distance-preserving maps.
  • The conformal automorphism group of invertible angle-preserving maps.
  • The diffeomorphism group of self-diffeomorphisms.
  • The homeomorphism group of self-homeomorphisms.

We can check these all act nonlinearly in one go: each action is transitive, hence there are group elements that send the origin to nonzero vectors.

The answers of Chappers, Nate and GoatsRule all fall under the second bullet point.