In this paper, a representation $\pi: G \to \operatorname{GL}(V)$ is said to be primitive if
- $\pi$ is irreducible and
- there exists no decomposition of $V$ as the direct sum of proper non-zero subspaces permuted by G.
Could you please describe what 2. technically means? It would be helpful if you could write the condition as a formula. Right now, this just sounds like 1. where there is no proper non-zero subspace which is invariant under $G$. I thought being permuted by $G$ and being invariant under $G$ is the same.
Thank you!
Consider the group of two by two matrices of the forms $$\pmatrix{a&0\\0&b}$$ and $$\pmatrix{0&a\\b&0}$$ for $ab\ne0$. Suppose the image of $\rho$ is this group. Then $\rho$ is irreducible, but is not primitive. The vector subspaces of the forms $\pmatrix{*\\0}$ and $\pmatrix{0\\*}$ form a direct sum composition of $K^2$ but are permuted by $G$.