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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?
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    $\begingroup$ I think it should be emphasised that in the complex case there is a huge literature. Going back to 1911, the second edition of Theory of groups of finite order by Burnside makes extensive use of character theory to prove results on permutation groups. For instance, there is a character theoretic proof that a permutation group of prime degree is either soluble or $2$-transitive. $\endgroup$
    – Mark Wildon
    Jun 9, 2019 at 20:22
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    $\begingroup$ Transitivity is equivalent to the trivial character appearing as a multiplicity one subrep. 2-transitivity is the same as the orthogonal complement of the trivial rep being irreducible over $\mathbb C$. 2-homogeneity is equivalent to the orthogonal complement of the trivial rep being irreducible over $\mathbb R$. $\endgroup$
    – Benjamin Steinberg
    Jun 9, 2019 at 22:29
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    $\begingroup$ The centralizer algebra is connected to the association scheme of the permutation group. $\endgroup$
    – Benjamin Steinberg
    Jun 9, 2019 at 22:30
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    $\begingroup$ After group completion and translation into algebraic topology, S_G (pt) for the Burnside ring vs K_G (pt) for the linear algebraic side (complex). See nLab. However, this isn't explicitly addressing the two example questions. $\endgroup$
    – AHusain
    Jun 9, 2019 at 23:48
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    $\begingroup$ @M.Winter : You're aware that there are several books with the title Permutation Groups? For example, Permutation Groups and Combinatorial Structures by Biggs and White? $\endgroup$
    – Timothy Chow
    Jun 10, 2019 at 2:43

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The comments give a reasonable selection of textbooks. I'll add a few more recent papers here, concentrating on the first part of the question. These generalize the three results mentioned by Benjamin Steinberg in the second comment above.

In Tsuzuku On multiple transitivity of permutation groups, Nagoya Math. J. 18 (1961), 93–109 there is a modern proof of a theorem of Frobenius that a permutation group $G \le S_n$ is $t$-transitive if and only if all irreducible characters $\chi^\lambda$ of $S_n$ with $n-\lambda_1 \le t/2$ restrict irreducibly to $G$. In particular, taking $\lambda = (n-r,r)$, this says that $G$ is $2r$-transitive only if $\chi^{(n-r,r)}$ restricts irreducibly to $G$.

The main theorem of Saxl Characters of multiply transitive groups, J. Alg. 34 (1975), 528–539 is the following refinement: Let $G$ be a permutation group of degree $n > 4r$. If the character $\chi^{(n-r,r)}$ restricts irreducibly to $G$ then either $G$ is $2r$-transitive or $r = 2$, $n = 9$ and $G = \mathrm{P\Gamma L}_2(\mathbb{F}_8)$.

Here $\mathrm{P\Gamma L}_2(\mathbb{F}_8)$ is the extension of $\mathrm{PGL}_2(\mathbb{F}_8)$ by the order $3$ Frobenius automorphism. This group is $3$-transitive but $4$-homogeneous. The permutation character of $S_9$ acting on the cosets of $\mathrm{P\Gamma L}_2(\mathbb{F}_8)$ is multiplicity-free. Such permutation characters of symmetric groups were partially classified by Saxl. The classification was completed by Godsil and Meagher, and independently, by me.

Peter M. Neumann also has many relevant papers: for example Generosity and characters of multiply transitive groups, Proc. Lond. Math. Soc. 31 (1975), 457–481.

Incidentally, all irreducible characters of symmetric groups are real (in fact rational) valued, and have the stronger property (see Geoff Robinson's comment) that they can be defined by homomorphisms into real general linear groups, as in the question.

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