My MMath student, Sam Hughes, recently submitted his dissertation entitled “Representation and character theory of the small Mathieu groups”; a copy is here.
The dissertation studies the ordinary (complex) character theory of M11 and M12; it includes the foundations of character theory, as well as details on how to construct M11 and M12 via the notion of “transitive extension”. I think Sam has done a beautiful job and should be congratulated!
We are in the process of writing up a paper including some of Sam’s results. In fact the paper comes from a slightly different point of view. Our main result is the following:
- If G is a sharply 5-transitive subgroup of Alt(12), then the character table of G is given by Table ***.
- If G is a sharply 4-transitive subgroup of Alt(11), then the character table of G is given by Table ***.
The point of this theorem is that we are able to construct the character table of G using only the assumption about multiple-transitivity – there is no direct reference to the Mathieu groups in this paper.
In the course of this research, I asked a question on MathOverflow here. Now seems a good time to thank the contributors to that discussion, especially Frieder Ladisch, for their help!