A finite group $\Gamma$ might be represented by a linear transformation

$$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$

or by permutations

$$\phi :\Gamma\to\mathrm{Sym}(n).$$

Of course, latter ones can be interpreted as linear representations with permutations matrices.

It appears to me that there are many interesting and non-trivial connections between these two types of representations. In particular, one seems to be able to derive properties for one of these by studying it from the perspective of the other.

Question:Is there any literature that explores these connections in detail? What are the relevant search terms?

I am especially interested in *real* representations, i.e., over $\Bbb R$. Here, I see many applications to geometry, e.g. symmetric polytopes, rigidity of symmetric frameworks, etc.

Here are examples of what I would consider interesting connections:

- Can transitivity, primitivity, 2-closedness or any other property of a permutation group nicely characterized in terms of its decomposition into irreducible real linear representations?
- Can a geometric property of a symmetric point arrangement (an orbit polytope, if you want) nicely characterized by a property of the induced permutation group on the points?

Permutation Groups? For example,Permutation Groups and Combinatorial Structuresby Biggs and White? $\endgroup$ – Timothy Chow Jun 10 '19 at 2:436more comments