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Recall that an element of a finite group is said to be real if it is conjugate to its inverse, and strongly real if the conjugating element can be chosen to be an involution.

Question: Is it true that for the 26 sporadic finite simple groups, all real elements of odd order are strongly real, apart from elements in the Atlas classes 3A and 5A of the McLaughlin simple group $M^cL$?

My question is motivated by this Mathoverflow question of A.Rupinski: Why are there so few quaternionic representations of simple groups ? . As noted there, $M^cL$ is the only sporadic finite simple group which has `quaternionic' representations. From the Atlas, the irreducible characters $\chi_{10}$ and $\chi_{13}$ of $M^cL$ each have Frobenius-Schur indicator $-1$.

The literature on `strongly real' finite simple groups usually looks at all conjugacy classes, not the classes of odd order elements.

I suspect that the number of (irreducible) quaternionic representations of a finite group is greater than or equal to the number of real conjugacy classes of odd order elements which are not strongly real. This might even be a known open conjecture.

PS

General discussion on relation between numbers of real/complex/quaternionic conjugacy classes and irreducible representations can be found here: MO46900: Are there “real” vs. “quaternionic” conjugacy classes in finite groups?

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    $\begingroup$ Nice question. Gow has a result in the spirit of your final paragraph: If the Sylow 2-subgroups of $G$ are dihedral or large enough semi-dihedral, then a quaternionic representation exists iff a non-strongly real but real element of odd order exists. See his 1979 J.Algebra paper. (Although I imagine you know of this already...) $\endgroup$
    – Nick Gill
    Nov 25, 2015 at 17:05
  • $\begingroup$ What is the obstacle to checking your question from the Atlas? $\endgroup$
    – Geoff Robinson
    Nov 25, 2015 at 18:43
  • $\begingroup$ Yes in principle the information is in the Atlas: for a given sporadic, check that all real 2-regular classes (bar the exceptions) show up in the square of some involution class sum. This can be done using the class multiplication formula. In practise one would probably use GAP. But it is alot of work, so I was hoping someone already knows the answer. $\endgroup$
    – John Murray
    Nov 25, 2015 at 20:54
  • $\begingroup$ I was thinking more of picking an element of odd order, and checking which maximal subgroups contain it. If it is strongly real, there must be a maximal subgroup in which it is still strongly real, and it may be more visible inside the maximal subgroup. I agree that checking class algebra constants might be a pain . $\endgroup$
    – Geoff Robinson
    Nov 25, 2015 at 21:47
  • $\begingroup$ I see three problems with checking maximal subgroups: (i) there are errors in the Atlas, some involving the maximal subgroups, (ii) ambiguity in identifying the fusion of conjugacy class from a maximal subgroup to the group, (iii) isoclinism - using the list of maximal subgroups in the -Atlas- the extended centralizer of the class 3A of $M^cL$ is isomorphic to $3_+^{1+4}:2S_5$. A Sylow 2-subgroup of this group is quaternion of order 16. So $2S_5$ is the non-split degree 2 extension of $2A_5$, and hence $3A$ is weakly real. But this is not evident from the -Atlas-. $\endgroup$
    – John Murray
    Nov 26, 2015 at 15:28

2 Answers 2

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If my coding is correct, then the answer to your question is Yes: All real elements of odd order in the sporadic simple groups are strongly real, with the exception of 3a, 5a in McL. With GAP, it takes only about a second to check the tables.

The following GAP function returns the class position of all strongly real classes in a character table:

StronglyRealClasses:= function( tbl )
    local kG,         # nr conjugacy classes
          invs,       # class positions of involutions
          nrinvs,     # nr of involutions
          sreals,     # class positions of strongly real classes
          prodcls,    # class positions in a product of 
                      # two involution class sums
          i, j, k;

    kG:= NrConjugacyClasses( tbl );

    invs:= Positions( OrdersClassRepresentatives( tbl ), 2 );
    nrinvs:= Length( invs );

    sreals:= [ ];

    for i in [ 1 .. nrinvs ] do 
    for j in [ i .. nrinvs ] do

        prodcls:= Filtered( [ 1..kG ], 
                    k-> ClassMultiplicationCoefficient(tbl, invs[i], invs[j], k) <> 0 
                    );
        UniteSet( sreals, prodcls );

    od; 
    od;
    return sreals;
end;

(This is quite naive. You could modify this function by omitting the inner for loop and replace invs[j] by invs[i], to make the function more efficient for your intended application of odd order strongly real elements, as suggested in your comment. But since it takes only about a second on my desktop computer to apply the above function to all the sporadic tables, I did not bother to rewrite it.)

The following function returns the class positions of all real elements of odd order, but not strongly real:

RCoOOnSR:= function( tbl )
    local rc, ooc, src;

    rc:= RealClasses( tbl );
    ooc:= PositionsProperty( OrdersClassRepresentatives( tbl ), IsOddInt );
    src:= StronglyRealClasses( tbl );

    return Difference( IntersectionSet( rc, ooc ), src );
end;

(Type ClassNames( tbl ){ RCoOOnSR( tbl ) }; to get names of the classes.)

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One more confirmation of OP's observation can be found in tables, from the paper:

Jordan Journal of Mathematics and Statisticscs (JJMS) l(2), 2008, pp. 97-103 97 STRONGLY REAL ELEMENTS IN SPORADIC GROUPS AND ALTERNATING GROUPS, IBRAHIM SULEIMAN

One can observe many real, but not strongly real classes, but most of them have even order, except group McL classes 3A, 5A - exactly as OP's proposal.

Table 3 will give the real classes which are not strongly real in Sporadic Groups

M11 all real elements are strongly real.
M12 all real elements are strongly real
M22 8A.
M23 8A
HS all real elements are strongly real.
J3 all real elements are strongly real.
M24 all real elements are strongly real.
McL 3A, 5A, 6A.
He all real elements are strongly real.
Ru all real elements are strongly real. 
Suz all real elements are strongly real.
ON all real elements are strongly real.
Co3 all real elements are strongly real.
Co2 16B.
Fi22 all real elements are strongly real.
HN 8A .
Ly all real elements are strongly real.
Th 8B.
Fi23 16AB, 22BC, 23AB.
Co1 all real elements are strongly real.
J4 all real elements are strongly real.
Fi_24 all real elements are strongly real.
BM all real elements are strongly real.
M 8C, 8F, 24F, 24G, 24H, 24J, 32A, 32B, 40A, 48A.

Table 4 will give the complete list for all non real elements in the Sporadic Groups

M11 8AB, 11AB.
M12 11AB.
J1 None
M22 7AB.
J2 None
M23 7AB, 11AB, 14AB, 15AB, 23AB.
HS 14AB.
J3 19AB.
M24 7AB, 14AB, 15AB, 21AB, 23AB.
McL 7AB, 9AB, 11AB, 14AB, 15AB, 30AB.
He 7AB, 7DE, 14AB, 14CD, 21CD, 28AB.
Ru 16AB.
ON 31AB.
Co3 11AB, 22AB, 20AB, 23AB.
Co2 14AB, 15BC, 23BC, 30BC.
Fi22 11AB, 16AB, 18AB, 22AB.
HN 19AB, 35AB, 40AB.
Ly 11AB, 22AB, 33AB. 
Fi23 16AB, 22BC, 23AB.
Co1 23AB, 39AB.
J4 7AB, 14AB, 14CD, 21AB, 28AB, 35AB, 42AB.
Fi_24 18GH, 23AB.
BM 23AB, 30GH, 31AB, 32CD, 46AB, 47AB.
 M 23AB, 31AB, 39CD, 40CD, 44AB, 46AB, 46CD, 47AB, 56BC, 59AB,
62AB, 69AB, 71AB, 78BC, 87AB, 88AB, 92AB, 93AB, 94AB, 95AB,
104AB, 119AB.
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