Timeline for Decomposition of induced representations / Refinement of Mackey's criterion
Current License: CC BY-SA 3.0
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| Feb 26, 2013 at 7:36 | comment | added | Jay Taylor | Just thought I'd add the following note: As an example of just how complicated this can become you might want to look at the article "Induced Cuspidal Representations and Generalised Hecke Rings" by Howlett and Lehrer. Even in this situation (which is reasonably optimal in the theory of finite reductive groups) they are left with an unknown 2-cocycle which one still has to compute to get an explicit description of the endomorphism algebra. | |
| Feb 26, 2013 at 5:52 | comment | added | Amritanshu Prasad | Dear Joël, thanks for your comment; I have edited my answer. The main thing to note is that the idea behind Mackey does not just give the dimension of the endomorphism algebra, but also the structure of the algebra; so if $H=1$, you don't just get the dimension of the regular representation, but actually its endomorphism algebra, from which you can (in principle) deduce the representation theory. | |
| Feb 26, 2013 at 5:45 | history | edited | Amritanshu Prasad | CC BY-SA 3.0 |
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| Feb 25, 2013 at 14:09 | comment | added | Joël | Dear Amritanshu, thanks for your answer. But Jim Humphreys' answer convinced me that you need to work more than "a little harder" to get that kind of information - as he mentions, if $H=1$, $W$ is the regular representation, and the method of Mackey gives us Dim $Hom_G(W,W)=|G|^2$, but one need extra (and somewhat harder) arguments to get the number of irreducible components in $W$, which certainly cannot be read on the decomposition of $Res_G^H W$, which is just a sum of $|G|$ trivial representation. Could you please explain, in a special case, of you envision this "work a little harder"? thx | |
| Feb 25, 2013 at 2:06 | history | answered | Amritanshu Prasad | CC BY-SA 3.0 |