Timeline for Which finite simple groups are rational-relative-real?
Current License: CC BY-SA 4.0
12 events
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| Jun 28, 2023 at 20:29 | comment | added | bl' | @LSpice: depends on how one defines "almost never happening". Section 7 of the article linked to in the answer rationality is also a rare phenomenon among solvable groups (compared to being RRR). | |
| Jun 28, 2023 at 13:30 | vote | accept | Theo Johnson-Freyd | ||
| Jun 28, 2023 at 11:09 | comment | added | Geoff Robinson | @LSpice : I think I was more restrained than that in my language in my answer to the previous question, but it is fair to say that symmetric groups have pretty small measure as a subset of the set of all finite groups. | |
| Jun 27, 2023 at 20:54 | comment | added | LSpice | It's a bit startling to hear a phenomenon (rationality) that happens for every symmetric group described as almost never happening! | |
| Jun 27, 2023 at 20:27 | answer | added | bl' | timeline score: 6 | |
| Jun 26, 2023 at 19:27 | comment | added | Geoff Robinson | I think a similar phenomenon occurs in ${\rm PSL}(2,p)$ when $p \equiv 1$ (mod $4$), for then $-1$ is a quadratic residue (mod $p$), and ${\rm PSL}(2,p)$ is not $RRR$ ( this also explains ${\rm Alt}(5)$ another way). | |
| Jun 26, 2023 at 19:17 | comment | added | Geoff Robinson | The values of an irreducible character of ${\rm Alt}(n)$ at an element of cycle-type $\lambda$ are all rational, unless the parts of $\lambda$ are all odd, and none are repeated, in which case, the character values lie in $\mathbb{Q}[ \sqrt{\epsilon \prod_{i} \lambda_{i}]},$ where $\lambda_{i}$ is the $i$-th part of $\lambda$, and the product is congruent to $\epsilon$ (mod $4$), This result appears in James and Kerber is used in a paper that Thompson and I wrote in 1994 in the Journal of Algebra, determining the field generated by all character values of a given alternating group. | |
| Jun 26, 2023 at 18:25 | comment | added | YCor | No, for $p$ prime, $\mathrm{Alt}(p)$ is not RRR if $p\equiv 1$ mod $4$, as $c^k$ is not conjugate to $c^{\pm 1}$ when $k$ is a multiplicative generator mod $p$ and $c$ is a $p$-cycle. (This is "visible" in $\mathrm{Alt}(5)$ viewed as group of motions of the icosahedron, since rotations of angle $2\pi/5$ and $4\pi/5$ are neither conjugate, nor inverse-conjugate.) | |
| Jun 26, 2023 at 16:58 | comment | added | Theo Johnson-Freyd | @DaveBenson Agreed. Perhaps I should ask: which groups fail this property? For example, I think alternating groups are not rational, but are RRR. I don't have enough intuition for the other series. | |
| Jun 26, 2023 at 16:53 | history | edited | Theo Johnson-Freyd | CC BY-SA 4.0 |
corrected a statement
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| Jun 26, 2023 at 14:42 | comment | added | Dave Benson | There are many finite simple groups with this property. For example, looking at the Atlas, in order of size, there are $L_2(7)$, $A_7$, $U_3(3)$, $M_{11}$, $A_8$, $M_{12}$, $U_3(5)$, $A_9$, $M_{22}$, and so on. You might try looking at the Atlas for yourself for the sporadic ones. | |
| Jun 26, 2023 at 14:16 | history | asked | Theo Johnson-Freyd | CC BY-SA 4.0 |