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Question Is true that if all real conjugacy classes of a finite group are strongly real, then all its real irreducible representations (irreps) are "strongly real"(symmetric) ? And vice verse ?

Definitions: Irrep is called real if its Frobenius Schur indicator is either +1 or -1, but not 0. Irrep is called "symmetric" ("orthogonal" ("strongly real")) if it has FS-indicator +1, which is equivalent that it is possible to realize it over real numbers.

The conjugacy class is called "real" if g-> g^{-1} preserves it. It is called "strongly real" if it is product of two involutions.

Context It is known that number of real conjugacy classes is the same as number of real irreps. A. Knutson's asked MO: Are there “real” vs. “quaternionic” conjugacy classes in finite groups?, i.e. whether it is possible to split "real" conjugacy classes in two subclasses such that they will correspond to splitting of irreps. Tim Dokchitser answered that no such splitting may exist in general which will be automorphism covariant....

The question above is weakening the A. Knutson's question.

Actually it has been asked explicitly in the paper Real Elements and Schur Indices of a Group, Amit Kulshrestha, Anupam Singh, arxiv 2011 04.

Where the authors present some positive evidences for the question above, however their examples do not cover GAP's SmallGroups List...

SubQuestion How difficult to check this in GAP ? (I just installed it, but never played).

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2 Answers

up vote 5 down vote accepted

The answer is No. My source is Rod Gow's rather wonderful little paper "Real-valued characters and the Schur index". Let me quote from the introduction:

It may happen that all elements of a group are strongly real while the group possesses real-valued characters of Schur index 2. An example is provided by the central product of quaternion and dihedral groups of order 8.

(Schur index equalling 2 is equivalent to Frobenius-Schur Indicator equalling -1.)

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On a related question, see our arXiv preprint, which also answers the converse question.

http://arxiv.org/abs/1210.3790

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Thank you very much ! –  Alexander Chervov Oct 24 '12 at 18:39
    
Welcome to Mathoverflow ! –  Alexander Chervov Oct 24 '12 at 18:40
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