We all know that a set of commuting diagonalizable matrices can be simultaneously put in diagonal form. My general question is:

Under what conditions can a set of (diagonalizable) matrices be simultaneously put in monomial form?

That is, when can find a basis with respect to which a collection of operators $\{A_i\}$ take the form $D_iP_i$ where $D_i$ is a diagonal matrix and $P_i$ is a permutation matrix? Clearly they must be diagonalizable, but I would like sufficient conditions.

For example:

Is it enough if

(a)there exist $n_i$ so that $A_i^{n_i}$ pairwise commute and are diagonalizable? Or(b)if the set $\{A_i\}$ is closed under conjugation, i.e. $A_iA_jA_i^{-1}=A_{i(j)}$ (which obviously implies(a))?

**Edit:** As Will Sawin points out, this is not enough as stated, so let me add another condition to **(b)**: suppose that $|\{A_i\}|\leq n$ where $A_i\in \mathbb{C}^{n\times n}$, so the number of generators is less than the dimension of the representation.

As you might guess this has something to do with representations of knot groups, but I am interesting in the general problem.