Timeline for Why is Klein's representation of $PSL_2(\mathbb{F}_7)$ hard to obtain?
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| Nov 12, 2014 at 23:04 | vote | accept | Reimundo Heluani | ||
| Nov 5, 2014 at 16:15 | answer | added | abx | timeline score: 17 | |
| Nov 5, 2014 at 14:25 | comment | added | Geoff Robinson | Just a tiny footnote to Frieder Ladisch's comment: whichever one of $\frac{q \pm 1}{2}$ is odd is the dimension of the irreducible representation of ${\rm PSL}(2,q)$, so when $q = 7$ that dimension is $3.$ The Weil representation of ${\rm SL}(2,q)$ has dimension $q$ and its two non-trivial irreducible constituents are the representations on the eigenspaces of the central involution. | |
| Nov 5, 2014 at 13:57 | comment | added | Frieder Ladisch | These representations arise as a constituent of the Weil representation of $\operatorname{SL}(2,q)$ ($q$ odd). The Weil rep. decomposes into two irreps of dims $(q\pm 1)/2$, where one irrep has $-I$ in the kernel and can thus be viewed as irrep of $\operatorname{PSL}(2,q)$. And the Weil rep. sends elements of order $4$ in $\operatorname{SL}(2,q)$ to the Fourier transform. See arxiv.org/abs/0903.1486 for some elementary proofs about the Weil rep in the finite case. | |
| Nov 5, 2014 at 13:21 | comment | added | Reimundo Heluani | @S.Carnahan the choice of $\sqrt{-7}$ corresponds to a labeling of one of the conjugation classes of $7$-cycles. The automorphism that David Speyer mentions should come from the outer automorphism of $G$ coming from $GL_2$. | |
| Nov 5, 2014 at 13:10 | comment | added | Geordie Williamson | Perhaps section 11.4.4 in "Representations of $SL_2(\mathbb{F}_q)$" by Cédric Bonnafé is interesting to you. He identifies $PSL_2(\mathbb{F}_7) \times \mathbb{Z}/2\mathbb{Z}$ as an exceptional complex reflection group of rank 3 ($G_{24}$ on the Shephard-Todd list). | |
| Nov 5, 2014 at 12:54 | comment | added | David E Speyer | @S.Carnahan A square root of $-7$, actually. One could imagine a combinatorial construction of the $6$ dimensional rep $W$ which is the direct sum of the two three dimensional reps and a combinatorial construction of an automorphism of $W$ with square $-7$. I don't know how to do it, though. | |
| Nov 5, 2014 at 12:49 | comment | added | Paul Broussous | Klein's representations seem to be the cuspidal representations of G (in the sense that they are not subrepresentations of parabolically induced representations). Parabolically induced representations have simple geometric models for they may be considered as spaces of functions on flag varieties. On the other hand, the cuspidal representations are the "difficult ones". They do not have simple geometric models. They are obtained in the $l$-adic cohomology of Deligne-Lusztig varieties. | |
| Nov 5, 2014 at 12:48 | comment | added | S. Carnahan♦ | Given that the representation is complex, any construction requires a choice of square root of minus one somewhere. I think this makes species-theoretic constructions difficult. | |
| Nov 5, 2014 at 12:35 | comment | added | Lev Borisov | If you look at weight two cusp forms for the principal congruence subgroup $\Gamma(7)$ of $SL_2(\mathbb Z)$, they form a dimension three representation. This seems geometric enough to me. | |
| Nov 5, 2014 at 9:57 | history | edited | Reimundo Heluani | CC BY-SA 3.0 |
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| Nov 5, 2014 at 9:47 | history | edited | Reimundo Heluani | CC BY-SA 3.0 |
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| Nov 5, 2014 at 9:40 | history | asked | Reimundo Heluani | CC BY-SA 3.0 |