A set of $d \times d$ real or complex matrices is commonly called irreducible if those matrices do not jointly preserve a linear subspace with dimension strictly between zero and $d$. A stronger hypothesis which is useful in multiplicative ergodic theory - for example, in Furstenberg's theorem on random matrix products - is that the matrices do not jointly preserve a finite union of nontrivial proper subspaces. This condition is referred to as strong irreducibility. Both of these properties also pass to the semigroup generated by the set. I would like to ask:
Is there a precise characterisation of sets (or semigroups) of matrices which are irreducible but not strongly irreducible?
An irreducible set of matrices which generates a finite semigroup will not be strongly irreducible. (Clearly if we can achieve this situation by multiplying each matrix by a nonzero scalar then the matrix set is also not strongly irreducible.) Another example of irreducibility without strong irreducibility is that in which $\mathbb{R}^d$ or $\mathbb{C}^d$ splits into a direct sum of $k$-dimensional subspaces which are permuted by the different matrices. Do these two mechanisms exhaust the range of possible examples?
Thanks in advance!