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I'm interested in the paper of Jan Saxl "The Complex Characters of the Symmetric Groups that Remain Irreducible in Subgroups".

I have only (not yet enough!) standard background on the representation theory of symmetric groups and theory of permutation groups.

But the paper seems to be involved and all the recent related works by Kleshchev seem to be on the case of positive characteristic. Thus I couldn't find recent thesis or books on Saxl's paper.

  1. So I hope someone can recommend me some papers or references to understand the paper of Jan Saxl. (I know only the basic theorems on the Specht modules and I have a few knowledge of the primitive subgroups of the symmetric groups.)

  2. Is there a modern reference for the Frobenius' paper and the theory of characters of multiply transitive permutation groups?

  3. I tried to read the paper of Frobenius, there he seemed to use the theory of substitution. Can you explain to me the definition of the substitution groups?

Thank you very much!

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  • $\begingroup$ This is a bit hard to answer because it is a little vague. For part 3, my recollection is that Frobenius says "substitution groups" where these days we would say "permutation groups". If I'm right (and maybe I'm not), then there isn't much more to be said... $\endgroup$
    – Nick Gill
    Commented Feb 2, 2021 at 16:45
  • $\begingroup$ @NickGill Thank you very much for your interest in my question! To be precise, I just want to know if there is a chapter of a book or some papers that cover all the theorems in the paper of Frobenius and so on. For example I want to know where can I find the proof of the statement that dimension of the representation corresponding to a partition $\lambda$ is the least integer $m$ for which $\chi^{\lambda}$ is a constituent of the permutation character of $S_{n}$ on the set of ordered $m$-tuples of points of $\{ n \}$. $\endgroup$
    – gualterio
    Commented Feb 3, 2021 at 7:36
  • $\begingroup$ @NickGill Actually I once have visited your website and I have been enjoying reading the thesis of Joanna Fawcett you've uploaded. Can you recommend any good reference where I can see any results on $\textbf{the restriction}$ of the representations of the whole permutation group to its various kind of subgroups? $\endgroup$
    – gualterio
    Commented Feb 3, 2021 at 7:40
  • $\begingroup$ @NickGille I'm afraid that large part of my question was relatively easy to answer. Since today, I began to understand earlier paper of Jan Saxl! But I hope to ask whether you could please recommend me some references for any well-established characterization of the representations of the primitive subgroups of the permutation groups. Thank you $\endgroup$
    – gualterio
    Commented Feb 3, 2021 at 16:32
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    $\begingroup$ You might find some of this answer of mine useful: mathoverflow.net/questions/333636/… $\endgroup$
    – Mark Wildon
    Commented Apr 2, 2021 at 19:38

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