Timeline for Finite simple groups and $ \operatorname{SU}_n $
Current License: CC BY-SA 4.0
12 events
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| Feb 2, 2022 at 19:58 | comment | added | David A. Craven | Hiss--Malle exclude $PSL_2(q)$ in their tables as there are too many of them. Their degrees are given elsewhere in the paper: $(q\pm1)/2$, $q$, $q\pm 1$ for $q$ odd, (with the obvious excluded two for $q$ even). For all finite subgroups you have to be a bit more careful, as there can be normalizers of $r$-subgroups that are not contained in the normalizer of a torus. You can already see this in dimension 3: $3^{1+2}:Q_8$ appears in Bray et al, in Table 8.5, in class $C_6$. These should be the only possible exceptions. (These are called extraspecial-type subgroups.) | |
| Feb 2, 2022 at 19:13 | comment | added | Ian Gershon Teixeira | For your remark about $ SU_{17} $ are you saying that every finite subgroup of $ SU_{17} $ is contained in a closed positive dimensional subgroup? Or just that every quasi simple finite subgroup is contained in a closed positive dimensional subgroup? | |
| Feb 2, 2022 at 18:26 | comment | added | Ian Gershon Teixeira | I do have a few questions about specifics. Looking at the table in Hiss-Malle I don't see $ PSL_2(11) $ in $ m=5,6 $ or $ PSL_2(19) $ in $ m=10,11 $ or $ PSL_2(23) $ in $ m=11,12 $ or $ PSL_2(9) $ in $ m=12 $ . Are the tables you used in Bray-Holt-Roney Dougal different somehow? Perhaps you could update your answer to explain how I can correctly read the tables? Also why does Hiss-Malle omit $ PSL_2(5)\cong A_5 $ for $ m=2 $? Is case $ m=2 $ considered too obvious to include? Also why doesn't $ PSL_2(7) $ show up in $ m=3 $? Is that simply because it is not irreducible? (that seems plausible) | |
| Feb 2, 2022 at 18:00 | comment | added | Ian Gershon Teixeira | Ok so I now understand your argument that my conjecture already fails in dimensional 17. Any rep of a finite simple group $ G $ into $ PSU_m $ comes from a rep of a quasi simple extension $ G' $ into $ U_m $. Moreover if the rep of $ G $ into $ PSU_m $ is maximal then it must be from an irrep $ G' $ in $ U_m $ but there are no irreps for $ m=17 $ because there are no irreps of any quasi simple groups with degree $ m=17 $. It suffices to check the paper Low dimensional rep of quasi simple groups by Hiss-Malle you recommended. This argument covers all simple groups not just linear and unitary. | |
| Feb 1, 2022 at 20:57 | comment | added | David A. Craven | Yes. You always have to look at central extensions. They give the centre in the tables. If the centre of the quasisimple group is not the centre of $SU$ then it is contained in the centralizer of a semisimple element and this is ignored. So these quasisimple groups are always simple mod $Z(SU)$, and are all such quasisimple groups. | |
| Feb 1, 2022 at 14:31 | comment | added | Ian Gershon Teixeira | wait but couldn't you have an irrep of a central extension of your simple group $ G $ and that would give a simple subgroup of $ PSU_m $ without giving an irrep of $ G $ of dimension $ m $? How do you rule that out? Do you have to look at dimensions of irreps for every central extension of $ G $ by a cyclic group? Is this possibility somehow already covered because your references also look at quasi simple groups? | |
| Feb 1, 2022 at 14:22 | comment | added | Ian Gershon Teixeira | Ok so an irrep in char $ p $ with $ p $ not dividing order of the group is same as char $ 0 $. And a rep of a simple group always has trivial kernel so it is an embedding. And again simple means the embedded subgroup is disjoint from the center of $ U_m $ so also embeds in $ PSU_m $. But a maximal closed subgroup is always image of an irrep so if you can find a dimension $ m$ such that no $ PSL_n(q) $ or $ PSU_n(q^2) $ has a (non modular) irrep of dimension $ m $ then it must be that $ PSU_m $ contains no $ PSL_n(q) $ or $ PSU_n(q^2) $subgroups. same for any simple | |
| Feb 1, 2022 at 14:00 | comment | added | David A. Craven | Yes, there were a few details hidden under the carpet here. Irreducible representations in characteristic 0 are the same as irreducible representations in characteristic $p$ for any $p$ not dividing the order of the group. So from these tables you can deduce what happens in char 0. I only used them as they are a handy reference. One can also use the tables in Hiss--Malle, which go up to dimension 250, to find degrees where there are no irreducible representations of linear/unitary groups of a given dimension either. Their tables explicitly list char 0. | |
| Feb 1, 2022 at 13:50 | comment | added | Ian Gershon Teixeira | I looked at the references and they seem to all be about maximal subgroups of other finite groups? (maybe I'm missing something?) How did you extract information about maximal subgroups of the compact real Lie group $ PSU_m $ based on these sources? Something about the fact that every complex rep of a finite group is unitary? By $ PSU_5 $ you mean the 24 real dimensional compact Lie group of $ 5 \times 5 $ unitary matrices with complex entries mod its center, right? | |
| Feb 1, 2022 at 0:43 | history | edited | David A. Craven | CC BY-SA 4.0 |
deleted 1 character in body
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| Jan 31, 2022 at 18:14 | vote | accept | Ian Gershon Teixeira | ||
| Jan 31, 2022 at 16:47 | history | answered | David A. Craven | CC BY-SA 4.0 |