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Is there a classification of such representations via unitriangular matrices over characteristic two fields?

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    $\begingroup$ At least for dihedral 2-groups, there is a large amount of relevant literature. I assume you want the classification of indecomposable representations in characteristic 2? Since the only irreducible representation of a 2-group is then the trivial one, you'd certainly expect unitriangular matrices. Classical papers (in the dihedral case) include C.M. Ringel, Math. Ann. 214 (1975) and V.M. Bonarendko, Math. Sb. 96 (1975). $\endgroup$ Oct 28, 2014 at 17:53
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    $\begingroup$ As Jim Humphreys says, (up to equivalence at least) the unitriangular condition is no restriction. Looking at indeccomposable representations (with noreal loss of generality) these two families of groups have tame representation type, which means that indecomposable representations can be reasonably parametrized, though there are infinitely many for each such group. $\endgroup$ Oct 28, 2014 at 18:08
  • $\begingroup$ Small typo due to rapid typing: the second author is Bondarenko. $\endgroup$ Oct 28, 2014 at 20:33
  • $\begingroup$ @JimHumphreys:Thanks for the references! I checked the paper of Bondarenko. Are those representations triangularizable? Can any representation be written as direct sum of indecomposable representations (in characteristic 2)? $\endgroup$
    – clyb
    Nov 3, 2014 at 19:17
  • $\begingroup$ @clyb: I've never looked closely at these groups relative to questions about splitting fields or Krull-Schmidt when you work in prime characteristic. But there shouldn't be any serious obstacle. (Aside from that, I don't know whether there is similar attention in the literature to classification of indecomposables for the quaternion group over these fields. But as Geoff remarks this is also a tame problem.) $\endgroup$ Nov 4, 2014 at 16:41

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