# Construction of representations of the Mathieu groups?

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:

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.

• In the case of $M_{12}$, I checked using GAP that every irreducible character except the two $16$-dimensional ones and the $54$-dimensional one is an irreducible factor, with multiplicity one, of some irreducible power $\bigwedge^i V$ ($0\leq i\leq 5$) of the standard $11$-dimensional representation $V$ (the nontrivial factor of the natural permutation representation). So, a posteriori, one can say that this provides a construction for all these representations, albeit not a very nice one. Sep 5, 2017 at 21:36
• Another ref: nickerson.org.uk/groups (construction of char 0 representations as permutation modules, used in linked Atlas of Finite Group Representations and GAP). Sep 6, 2017 at 3:51
• @Gro-Tsen Sorry, I do not quite understand "irreducible factor" - over "C" it should be just subrepresentation. And what means multiplicity one ? Any way - may be you can post your GAP code and some extended comments in an answer ? That would be quite useful (at least for me). Sep 6, 2017 at 20:54

(This is not really an answer — or merely a very partial one —, but a clarification of the comment I posted earlier, as per OP's request.)

Consider the case of $M_{12}$. Let $V$ be its ($11$-dimensional) "standard" representation, by which I mean the nontrivial factor of $\mathbb{C}^{12}$ on which $M_{12}$ acts by permutation of coordinates (in other words, $V$ is the set of elements of $\mathbb{C}^{12}$ of zero sum). A simple computation using GAP (see below) reveals that, among the exterior powers $\bigwedge^i V$ for $0\leq i\leq 5$, (A) every irreducible factor occurs with multiplicity one (by this I mean that in the decomposition of $\bigwedge^i V$ into irreducible components, they are pairwise non-isomorphic), and (B) among such factors appear $12$ of the $15$ irreducible representations of $M_{12}$, namely all except the two $16$-dimensional ones and the $54$-dimensional one.

Furthermore recall the following standard and easy fact (see, e.g., Fulton & Harris, Representation Theory (2004), §2.4, formula (2.31)): if $U$ is a (finite dimensional) representation of a finite group $G$, and $W$ is an irreducible representation of $G$ with character $\chi_W$, then the projection of $U$ onto the sum of all its irreducible factors isomorphic to $W$ is given by $\psi(x) = \frac{\dim W}{\#G} \sum_{g\in G} \overline{\chi_W(g)}\cdot gx$. Consequently, if $U$ is known (say, $\bigwedge^i V$ in the context above) and $W$ is an irreducible representation which is to be constructed but whose character $\chi_W$ is known, and which occurs with multiplicity exactly $1$ in $U$, we can indeed construct $W$ as the set of $x \in U$ such that $\sum_{g\in G} \overline{\chi_W(g)}\cdot gx = \frac{\#G}{\chi_W(1)}\, x$.

For example, $\bigwedge^3 V$ (which is completely explicit) is the direct sum of the $45$-dimensional and $120$-dimensional representations: this provides a construction of the former, say, as the set of $x \in \bigwedge^3 V$ such that $45x + 5 \sum_{g\in\mathrm{2A}} gx - 3 \sum_{g\in\mathrm{2B}} gx + \cdots + \sum_{g\in\mathrm{11B}} gx = 2112\, x$ (labeling the classes as in GAP and the ATLAS): not a very nice construction, but still arguably a form of answer. Of course, one can do better by looking more closely at the values in the caracter table, and discriminate on the value of just about any single class: for example, the same $45$-dimensional representations is the set of $x \in \bigwedge^3 V$ such that $\sum_{g\in\mathrm{2A}} gx = 44x$ (whereas the $120$-dimensional one is the set of $x \in \bigwedge^3 V$ such that $\sum_{g\in\mathrm{2A}} gx = 0$), which is already a bit nicer.

So, if you consider this a construction, this constructs $12$ of the $15$ irreducible representations of $M_{12}$.

GAP session illustrating the computation:

 ┌───────┐   GAP, Version 4.7.9 of 29-Nov-2015 (free software, GPL)
│  GAP  │   http://www.gap-system.org
└───────┘   Architecture: x86_64-pc-linux-gnu-gcc-default64
Components: small 2.1, small2 2.0, id2 3.0, trans 1.0, prim 2.1
Packages:   CTblLib 1.2.2, FGA 1.2.0, GAPDoc 1.5.1
gap> m12 := MathieuGroup(12);
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6), (1,12)(2,11)(3,6)
(4,8)(5,9)(7,10) ])
gap> ctb := CharacterTable("M12");
CharacterTable( "M12" )
gap> ccl := ConjugacyClasses(m12);;
gap> ctb := CharacterTableWithStoredGroup(m12,ctb);
CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)
(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) )
gap> Display(ctb);
M12

2   6  4  6  1  2  5  5  1  2  1  3  3   1   .   .
3   3  1  1  3  2  .  .  .  1  1  .  .   .   .   .
5   1  1  .  .  .  .  .  1  .  .  .  .   1   .   .
11   1  .  .  .  .  .  .  .  .  .  .  .   .   1   1

1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a 11a 11b
2P  1a 1a 1a 3a 3b 2b 2b 5a 3b 3a 4a 4b  5a 11b 11a
3P  1a 2a 2b 1a 1a 4a 4b 5a 2a 2b 8a 8b 10a 11a 11b
5P  1a 2a 2b 3a 3b 4a 4b 1a 6a 6b 8a 8b  2a 11a 11b
11P  1a 2a 2b 3a 3b 4a 4b 5a 6a 6b 8a 8b 10a  1a  1a

X.1       1  1  1  1  1  1  1  1  1  1  1  1   1   1   1
X.2      11 -1  3  2 -1 -1  3  1 -1  . -1  1  -1   .   .
X.3      11 -1  3  2 -1  3 -1  1 -1  .  1 -1  -1   .   .
X.4      16  4  . -2  1  .  .  1  1  .  .  .  -1   A  /A
X.5      16  4  . -2  1  .  .  1  1  .  .  .  -1  /A   A
X.6      45  5 -3  .  3  1  1  . -1  . -1 -1   .   1   1
X.7      54  6  6  .  .  2  2 -1  .  .  .  .   1  -1  -1
X.8      55 -5  7  1  1 -1 -1  .  1  1 -1 -1   .   .   .
X.9      55 -5 -1  1  1  3 -1  .  1 -1 -1  1   .   .   .
X.10     55 -5 -1  1  1 -1  3  .  1 -1  1 -1   .   .   .
X.11     66  6  2  3  . -2 -2  1  . -1  .  .   1   .   .
X.12     99 -1  3  .  3 -1 -1 -1 -1  .  1  1  -1   .   .
X.13    120  . -8  3  .  .  .  .  .  1  .  .   .  -1  -1
X.14    144  4  .  . -3  .  . -1  1  .  .  .  -1   1   1
X.15    176 -4  . -4 -1  .  .  1 -1  .  .  .   1   .   .

A = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
= (-1+Sqrt(-11))/2 = b11
gap> # std is the character of the "standard" representation:
gap> std := Irr(ctb)[2];
Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)
(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),
[ 11, -1, 3, 2, -1, -1, 3, 1, -1, 0, -1, 1, -1, 0, 0 ] )
gap> # stdalt is the list of exterior powers of std
gap> stdalt := List([1..5], i->AntiSymmetricParts(ctb,[std],i)[1]);
[ Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)
(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),
[ 11, -1, 3, 2, -1, -1, 3, 1, -1, 0, -1, 1, -1, 0, 0 ] ),
Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)
(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),
[ 55, -5, -1, 1, 1, -1, 3, 0, 1, -1, 1, -1, 0, 0, 0 ] ),
Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)
(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),
[ 165, 5, -11, 3, 3, 1, 1, 0, -1, 1, -1, -1, 0, 0, 0 ] ),
Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)
(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),
[ 330, 10, -6, 6, -3, -2, -2, 0, 1, 0, 0, 0, 0, 0, 0 ] ),
Character( CharacterTable( Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)
(4,10,5,6), (1,12)(2,11)(3,6)(4,8)(5,9)(7,10) ]) ),
[ 462, -10, 14, 3, 3, 2, -6, 2, -1, -1, 0, 0, 0, 0, 0 ] ) ]
gap> # Compute the multiplicities of the irreducibles in each exterior power:
gap> List([1..5], i->List([1..Length(Irr(ctb))], j->ScalarProduct(stdalt[i],Irr(ctb)[j])));
[ [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0 ],
[ 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0 ],
[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0 ],
[ 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 1 ] ]

• Curiously enough, for $M_{24}$, we also get every irreducible representation but three as a multiplicity one irreducible factor of an exterior power $\bigwedge^i V$ of the standard representation $V$ (we get all but the irreducibles with dimensions $1265$, $5313$ and $10395$); but this time it is no longer the case that every irreductible factor of $\bigwedge^i V$ has multiplicity one. Sep 7, 2017 at 14:26

I would like to draw your attention to an article by Nick Gill and (his then MMath student) Sam Hughes The character table of a sharply 5-transitive subgroup of $$A_{12}$$, that constructs the character table of the Mathieu group $$M_{12}$$ assuming only the properties that are mentioned in the title.

Not really an answer to the question, but check out the very cool Bachelor's Thesis by A. Joshi (from IIT Madras).