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asked | Outer group automorphisms preserving conjugacy classes of pairs of commuting elements |
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comment |
Outer automorphisms of Borel subgroup
Thanks, Aakumadula ! Probably one can deduce that $\theta$ (modulo an inner automorphism) respects root subgroups by using that normal abelian subgroups of $B$ are in bijection with positive roots (at least for the type $A_n$). In particular, maximal normal abelian subgroups of $B$ correspond to simple roots. The automorphism $\sigma'$ of $B$ given by composition of taking transpose w.r.t (1n) -- (n1) diagonal and taking the inverse. Does this coincide with your $\sigma$? (it corresponds to the symmetry of the $A_n$ Dynkin diagram). |
Mar
5 |
comment |
Outer automorphisms of Borel subgroup
Thanks! Sure $B$ coincides with its normalizer in $GL(n, F)$. But it appears that there are non-trivial outer automorphisms of $B$ , e.g., coming from automorphisms of $F$. Also, in this paper deepblue.lib.umich.edu/handle/2027.42/30473 the group $Out(B)$ is computed when $F$ is the field of $2$ elements. It is surprisingly large (in this case, of course, $B$ is the group of unipotent upper-triangular matrices) |
Mar
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revised |
Outer automorphisms of Borel subgroup
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Mar
5 |
revised |
Outer automorphisms of Borel subgroup
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5 |
comment |
Outer automorphisms of Borel subgroup
Hi Jon! nice to see you here. |
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28 |
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Non-symmetric Braiding on finite group Representation Categories
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28 |
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Non-symmetric Braiding on finite group Representation Categories
Victor is right. I will edit the above answer. |
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