The Mathieu groups are beautiful simple finite groups. (They were the first sporadic groups to be discovered in 1861-1870). They are related with many other miraculous constructions in mathematics: Golay error-correcting codes, Steiner systems, K3 surfaces and moonshine, etc...

One might expect that construction of irreducible representations over complex numbers of such distinguished groups should also be beautiful, however googling I was unable to find something like that.

**Question** What are constructions (hopefully "nice") of irreducible representations of the Mathieu groups ?

Googling suggests:

- Frobenius found character tables of the Mathieu groups in 1901 (I do not have reference).
- G. James found all modular irreps in The modular characters of the Mathieu groups 1973, but his methods are quite specific to modular case
- J.F Humphreys in 1980 considered more general questions of projective characters and character for automorphism group. The projective characters of the Mathieu group M12 and of its automorphism group, but there is no free access to that paper ...

Some simple considerations leads to the following observations: for example M12 acts on 12 points, hence one has 12-dim permutation representation, it is natural to expect that 11-dim subrepresentation is irreducible and it is restriction of the one of from symmetric group S12 or Alternating groups A12. Mathieu group M12 also has 55-dim irrep it is natural to guess that it is wedge-square of 11-dimensional. One can also see that dimensions of irreps coinciding in M12 and S12 and A12 are: 1, 11,54,55. So 54-dim irrep of Mathieu probably is restriction of the one for S12/A12. (What is this representation for S12/A12 ? )

Character table for e.g. M12 can be obtained by MAGMA http://magma.maths.usyd.edu.au/calc/ for free:

```
load m12;
CharacterTable(G);
```

For A12:

```
AlternatingCharacterTable(12)
```

Further info on e.g. M12 irreps can be found here: https://groupprops.subwiki.org/wiki/Linear_representation_theory_of_Mathieu_group:M12, e.g. dimensions of irreps: 1,11,11,16,16,45,54,55,55,55,66,99,120,144,176

Information of complex/real representations: MO Strongly real elements of odd order in sporadic finite simple groups, some discussion: MO Atlas of finite groups, Character table of automorphism group of sporadic group.

A part of motivations comes from: MO Monstrous Langlands-McKay ... , the other part from the discussion with S. Galkin, who found in 2010 G-FANO THREEFOLDS ARE MIRROR-MODULAR that Gromov-Witten invariants of certain Fano 3-folds can be expressed via $\eta$-producs related to the Mathieu group M24 by the construction of G.Mason, extending V.Golyshev's results on Fano 3-folds and Moonshine.

a posteriori, one can say that this provides a construction for all these representations, albeit not a very nice one. $\endgroup$Atlas of Finite Group Representationsand GAP). $\endgroup$