Talk:Brauer's theorem on induced characters
Latest comment: 16 years ago by Messagetolove
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Is this related to Brauer's three main theorems? Geometry guy 19:23, 13 May 2007 (UTC)
- Not directly. Brauer's three main theorems are theorems from modular representation theory. The first main theorem says that whenever D is a p-subgroup of a finite group G, there is a bijection between p-blocks of G with defect group D and p-blocks of the normalizer in G of D with defect group D. The second main theorem is too technical to state fully here, but is about generalized decomposition numbers and Brauer correspondent blocks. The third main theorem says that under mild assumptions, in Brauer correspondence ( which probably should have its own article) only the principal block corresponds to the principal block. Both the second and third main theorems can be used to obtain extra orthogonality relations for group characters which can be stated purely in terms of ordinary characters, but for which there does not presently seem to be a proof purely in terms of ordinary characters. However, Brauer's induction theorem is a statement about how to generate the ordinary (integral) character ring, and does not require any modular representation theory to prove it (although some of the ideas of some proofs are not dissimilar in spirit from some of the ideas which occur in Brauer's modular work).
- In fact, there are some applications of Brauer's Second Main Theorem which are used to provide blockwise refinements (loosely speaking) of Brauer's characterization of characters in much more recent work of mathematicians such as Michel Broue and Lluis Puig. Messagetolove 23:26, 13 May 2007 (UTC)
Many thanks!! I suppose my real question was: is it worth adding appropriate links between these articles? If so, please feel free to add such links! Geometry guy 00:15, 14 May 2007 (UTC)
- You are welcome. The articles to do with Brauer and those to do with Modular representation theory could all do with quite a bit of work. I have had some attempts at improvement, but would need quite a bit of time to do a really thorough job.
Messagetolove 07:22, 14 May 2007 (UTC)