Timeline for Representations of the orthogonal group O(n) vs representations of the special orthogonal group SO(n), over an arbitrary field
Current License: CC BY-SA 3.0
9 events
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Aug 15, 2013 at 7:50 | comment | added | Communicative Algebra | @JimHumphreys: Serre's Groupes de Grothendieck des schémas en groupes réductifs déployés and/or Jantzen tell me that for a split reductive group scheme like SO(n), the simple representations defined over k remain simple over the algebraic closure, and conversely all simple representations over the algebraic closure are already defined over k. But in general, this is not true, and I don't think either of Serre or Jantzen use the term “reductive” to include non-connected groups like O(n). | |
| Aug 15, 2013 at 7:48 | comment | added | Communicative Algebra | @JimHumphreys: I’m not sure what you mean by “representations will be studied over an algebraically closed field”—I want the representations to be defined over the same field as the group itself. Do I need to make this more precise in the question? | |
| Aug 14, 2013 at 19:00 | comment | added | Jim Humphreys | P.S. As pointed out in a comment, the Springer GTM 255 by Goodman-Wallach is likely to be a reliable source for your purpose. I don't have that 2009 edition, but the first edition titled Representations and Invariants of the Classical Groups (Cambridge, 1998) has the relevant material in sections 5.2.2 and 10.2.5. | |
| Aug 14, 2013 at 17:34 | comment | added | Jim Humphreys | As long as the characteristic of the field of definition is good (not 2), as you assume, it doesn't seem to matter over which field the groups are defined and split. Representations will be studied over an algebraically closed field, where for $n$ even the methods will rely on standard induction/restriction. Older group representation texts for physicists probably cover orthogonal groups, while Jantzen's book on algebraic groups may be overkill. | |
| Aug 14, 2013 at 13:35 | history | edited | Communicative Algebra | CC BY-SA 3.0 |
added: summary of the statements I am looking for
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| Aug 13, 2013 at 22:17 | comment | added | emiliocba | For $k=\mathbb C$, it's explained in Section 5.5.5 of Goodman-Wallach's book "Symmetry, represenations, and invariants". | |
| Aug 7, 2013 at 7:49 | comment | added | Communicative Algebra | For the time being, the “fair amount of detail” I am referring to may be found in Proposition 3.18 and in Section 4.2 of arXiv:1308.0796. | |
| Jul 27, 2013 at 10:50 | history | asked | Communicative Algebra | CC BY-SA 3.0 |