Timeline for Which groups have only real and quaternionic irreducible representations?
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| Jul 7, 2020 at 23:38 | comment | added | Torsten Schoeneberg | @Skip: I know it's almost ten years since you wrote this, but I just came across it and wonder if these contradictory definitions are the clue to explaining the contradiction between the accepted answer and my answer to math.stackexchange.com/q/310678/96384. Would you mind having a look at those, as well as my recent answer to math.stackexchange.com/q/3738143/96384 which talks about when certain spin representations are complex/quaternionic/real? | |
| Apr 13, 2017 at 12:57 | history | edited | CommunityBot |
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| Nov 29, 2010 at 16:03 | comment | added | Skip | For future readers of this list of comments, here are some counterexamples to the equivalence if the hypotheses are relaxed. If $G$ is not compact, the spin reps of Spin(5,3) are complex in my sense -- meaning not defined over $\mathbb{R}$ -- but have an invariant symmetric bilinear form. If the representation is not irreducible: take $G$ compact and $V$ an irreducible representation that is complex in my sense, so $V^*$ is the complex conjugate of $V$ and is not $V$. Then $V \oplus V^*$ is both real and quaternionic in Baez's sense but is only real in my sense ("no quaternionic structure"). | |
| Nov 28, 2010 at 17:00 | comment | added | José Figueroa-O'Farrill | Yes, sorry, perhaps I should have said explicitly in my comment that this is of course only true in the context of unitary irreps. | |
| Nov 28, 2010 at 2:50 | comment | added | Skip | The section in Bröcker and tom Dieck that you point to contained the definition I had in mind (and expressed poorly) -- this seems like the natural definition to me. Now that you mention it, I see that their/my definition agrees with John's in the very special case where the group is compact and the representation is irreducible. (If we allow, say $V \oplus V^*$, then you can always endow that with a nondegenerate symmetric or skew-symmetric bilinear form.) So in the strict sense of the question, I stand corrected, the two definitions agree. | |
| Nov 27, 2010 at 18:26 | comment | added | José Figueroa-O'Farrill | I believe that the usage of real/complex/quaternionic both in John's question and in Noah's answer are only superficially different. In fact, the relation between the two is explained, for instance, in §II.6 of the book Representation theory of compact Lie groups by Bröcker and tom Dieck. For example, let $U$ be an irreducible real representation. If $U$ is real (resp. quaternionic) in the sense of this answer (and Noah's), if and only if its complexification is of real (resp. quaternionic) type. | |
| Nov 27, 2010 at 16:35 | history | answered | Skip | CC BY-SA 2.5 |