Timeline for Failure of a basic fact from Representation Theory
Current License: CC BY-SA 3.0
9 events
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| Feb 9, 2020 at 14:06 | comment | added | Bernie | Fantastic answer! I learned a lot | |
| Aug 20, 2012 at 0:46 | comment | added | Jim Humphreys |
P.S. I should add that the most direct construction for $SL_2$ (say over any field of char 7) just uses the spaces of homogeneous polynomials in two variables of degrees 0, 1, ..., 6, of respective dimensions 1, 2, ..., 7. Here -I acts trivially in dim 1, 3, 5, 7 and gives the desired irreducibles for $PSL_2$. Also, I should add that the group over the prime field is often called $L_2(7)$ but is isomorphic to $L_3(2)$ (the name used by Conway et al. in their book An Atlas of Brauer Characters).
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| Aug 19, 2012 at 23:58 | comment | added | Jim Humphreys | @Alexander: For this group in characteristic 7 (but not over the complex field), the irreducibles have dimensions 1, 3, 5, 7, and arise from classical Lie theory combined with reduction mod 7. This starts with the usual Lie algebra matrix representations for the special linear case but then involves an integral (say Chevalley) lattice and reduction mod 7. Since 7 divides the group order, the ordinary and 7-modular theories diverge to some extent but are then related by Brauer theory. | |
| Aug 18, 2012 at 18:28 | comment | added | Alexander Chervov | I mean psl (2,7)... | |
| Aug 18, 2012 at 18:27 | comment | added | Alexander Chervov | How to describe 5dim irrep of pal(2 7) ? I an playing with this group now. Actually over complex but would be happy to know more. | |
| Aug 7, 2012 at 23:41 | vote | accept | ARupinski | ||
| Aug 7, 2012 at 17:01 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 7 characters in body
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| Aug 7, 2012 at 11:52 | history | edited | Jim Humphreys | CC BY-SA 3.0 |
added 1082 characters in body
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| Aug 7, 2012 at 0:35 | history | answered | Jim Humphreys | CC BY-SA 3.0 |