Timeline for Irreducible representations of product of profinite groups
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 15, 2021 at 19:51 | comment | added | Benjamin Steinberg | Since he said irreducible you can automatically assume finite dimensional if the representation is on a Hilbert space. | |
| Jul 15, 2021 at 19:49 | comment | added | Alex B. | @BenjaminSteinberg Oh yeah, sorry, I was not disputing that. Now corrected -- thanks! | |
| Jul 15, 2021 at 19:48 | history | edited | Alex B. | CC BY-SA 4.0 |
inserted the hypothesis that the representations be finite dimensional
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| Jul 15, 2021 at 16:34 | comment | added | Benjamin Steinberg | Yes. The OP should have been clearer. As you point out even for finite groups you need a splitting field. But it is important that reps are finite dimensional to get it to factor through a finite quotient | |
| Jul 15, 2021 at 15:45 | vote | accept | Martin Skilleter | ||
| Jul 15, 2021 at 14:55 | comment | added | Alex B. | @BenjaminSteinberg Yes, we posted more or less simultaneously. I did not take the "topological vector space" part of the question to implicitly assume that we are working over the complex numbers. For example number theorists routinely consider continuous representations of profinite groups (e.g. of absolute Galois groups) over all sorts of rings. | |
| Jul 15, 2021 at 14:44 | comment | added | Benjamin Steinberg | Anyway +1. The question could be more specific on the hypotheses | |
| Jul 15, 2021 at 14:02 | comment | added | Benjamin Steinberg | The complex part is essentially my answer in the comments except you need to use that all irreducible representations are finite dimensional since the regular representation of a compact group is faithful. I think it is implicit from the question mentioning a counting argument for finite groups and talking about topological vector space and unitary representations that the OP was working over the complex numbers | |
| Jul 15, 2021 at 13:16 | history | answered | Alex B. | CC BY-SA 4.0 |