Timeline for Is there a standard name for this index 2 subgroup in an affine group over a finite field of odd char?
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| Sep 19, 2011 at 22:20 | comment | added | Yemon Choi | As to where the group arises: nothing so interesting as the connections mentioned by Chris Godsil and Noam Elkies, I'm afraid. We are looking at a certain constant associated to any finite group - specifically, the l^1-norm of the "diagonal idempotent" associated to the centre of the complex group algebra - and can obtain a precise formula when the group is Frobenius with abelian kernel and complement. For the full ax+b group over ${\bf F}_q}$ the constant is $\leq 5$, while for the $a^2x+b$ subgroup the constant turns out to be $O(q)$. | |
| Sep 19, 2011 at 22:13 | comment | added | Yemon Choi | Thanks for the comments - I guess "a^2x+b$ group would work best for us, since the likely readers of the eventual paper will be abstract harmonic analysis people, who are comfortable with talk of the "ax+b" group over various fields. | |
| Sep 18, 2011 at 0:23 | comment | added | Noam D. Elkies | Paley graphs must have $q \equiv 1 \bmod 4$ (though there's a directed version for $q \equiv -1 \bmod 4$).$$ $$ For $q$ prime, this group is also the Galois group of the cover of modular curves ${\rm X}(q) / {\rm X}_0(q)$ over ${\bf C}$. [This fits in with the ${\rm PSL}_2({\bf F}_q)$ picture.] For higher prime powers there's a similar description using Shimura curves.$$ $$ Given Y.Choi's research interests I wonder where this group arises; neither Paley graphs nor modular curves seems a likely context. | |
| Sep 17, 2011 at 21:41 | comment | added | Chris Godsil | This group arises as a large subgroup of the automorphism group of a Paley graph. (It's not the full automorphism group, you have to add the field automorphisms to get that.) In any case it appears in many papers and I have never seen it given a name. And $G_q$ is used for so many things in group theory, your contribution to the pile is unlikely to be remarked on. | |
| Sep 17, 2011 at 5:18 | comment | added | Noam D. Elkies | ...so if you need a symbol for it, probably $B$ (for Borel) or $B_q$. | |
| Sep 17, 2011 at 4:47 | comment | added | Noam D. Elkies | Either "the Borel subgroup of ${\rm PSL}_2({\bf F}_q)$", or "the $a^2 x + b$ group" — the latter because it acts on the affine line by permutations of the form $x \mapsto a^2 x + b$ for $a \in {\bf F}_q^*$ and $b \in {\bf F}_q$. | |
| Sep 17, 2011 at 4:33 | history | asked | Yemon Choi | CC BY-SA 3.0 |