Timeline for Monstrous Moonshine for Thompson group $Th$?
Current License: CC BY-SA 3.0
31 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2016 at 10:05 | answer | added | S. Carnahan♦ | timeline score: 5 | |
| May 5, 2015 at 17:10 | vote | accept | Tito Piezas III | ||
| May 5, 2015 at 2:02 | answer | added | BRayhaun | timeline score: 7 | |
| Mar 3, 2014 at 18:42 | comment | added | Tito Piezas III | @S.Carnahan I thought as much. I looked at my copy of the C-N paper and there it was. | |
| Mar 3, 2014 at 14:32 | comment | added | S. Carnahan♦ | The centralizer of 6B is $6.Sz$. This can be found in Table 2a of the original Conway-Norton paper. | |
| Feb 26, 2014 at 5:17 | comment | added | Tito Piezas III | @JeffHarvey Thanks. By the way, do you know the McKay-Thompson series giving rise to moonshine for the Suzuki group? Excluding the pariahs, Janko groups, and Mathieu groups, this is the only one that seems hard to find and not in this list. (But I think its $T_{6B}$.) | |
| Feb 25, 2014 at 16:51 | comment | added | Jeff Harvey | @TitoPiezasIII The other functions for $E_k$, $k=6,8,10,14$ are discussed in Borcherds paper "Automorphic forms on $O_{s+2,2}(R)$ and infinite products". In example 2 of sec. 15 he give the weight $1/2$ form whose lift is $E_6$ and then the required functions for $E_8,E_{10},E_{14}$ follow from the fact that $E_8=E_4^2$, $E_{10}=E_4 E_6$ and $E_{14}= E_4^2 E_6$. | |
| Feb 23, 2014 at 23:52 | comment | added | S. Carnahan♦ | @JeffHarvey I think so, although I am rather uncertain about the properties of unlifted coefficients of reverse lifts. | |
| Feb 23, 2014 at 23:50 | comment | added | Jeff Harvey | @S.Carnahan So $j^{1/3}$ is the lift of $b-4 \vartheta(\tau)$ and each term in $b-4 \vartheta(\tau)$ can be written as a sum of (virtual) Thompson irreps. | |
| Feb 23, 2014 at 23:10 | comment | added | S. Carnahan♦ | @JeffHarvey I don't see an obvious connection, but the fact that $j^{1/3} = E_4/\eta^8$ suggests that a relation may exist. | |
| Feb 23, 2014 at 22:11 | comment | added | Jeff Harvey | @S.Carnahan Do you see a relation between the moonshine for the Thompson group exhibited in Generalized Moonshine or in the work of Griess and Lam and the apparent moonshine exhibited here for the weight 1/2 modular form $b(\tau)$? | |
| Feb 23, 2014 at 17:08 | comment | added | Tito Piezas III | @S.Carnahan I've added a link to the post for a list of moonshine for other groups, en.wikipedia.org/wiki/Monstrous_moonshine#Generalized_Moonshine | |
| Feb 23, 2014 at 17:04 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Added after-note.
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| Feb 21, 2014 at 13:40 | comment | added | Jeff Harvey | @TitoPiezasIII Borcherds defines $b(\tau)$ this way so that its Borcherds lift gives $E_4$. But the weight 12 discriminant function $\Delta(\tau)$ is the Borcherds lift of $12 \vartheta(\tau)$ so adding $120 \vartheta(\tau)$ gives a weight 1/2 modular form with Borcherds lift $E_4 \Delta^{10}$ and indeed one has the q expansion $E_4 \Delta^{10}=q^{10}-27000 q^{12}+4096000 q^{13} + \cdots$ exhibiting dimensions of Thompson irreps. However this lift only involves the coefficients $b(n^2)$ in the q expansion of $b$ while you see moonshine in $b(n)$ for n square free. | |
| Feb 20, 2014 at 21:15 | comment | added | S. Carnahan♦ | @F.C. The modular moonshine puzzle was solved by Griess and Lam last year: arxiv.org/abs/1308.2270 . As it happens, there is already a moonshine for the Thompson group, via the function $j(\tau/3)^{1/3} = q^{-1/9} + 248q^{2/9} + 4124q^{5/9} + \cdots$. This function is the character of the 3C-twisted module of $V^\natural$, and has a natural $Th$-representation structure. | |
| Feb 20, 2014 at 15:04 | comment | added | F. C. | There are a few words at the end of section 4 of Borcherds' article "Modular Moonshine II" (math.berkeley.edu/~reb/papers/modular2/modular2.pdf) about $E_8(3)$ and the Thompson group. But this concerns modular representations of Th. | |
| Feb 20, 2014 at 14:08 | comment | added | Jeff Harvey | The Thompson group is a subgroup of the Chevalley group $E_8(3)$ and the Eisenstein series $E_4$, which is equal to the $E_8$ theta function is the Borcherds lift of the weight $1/2$ modular form you wrote down. Perhaps this is a starting point for trying to understand the connection between Thompson irreps and $b(\tau)$. | |
| Feb 20, 2014 at 13:32 | comment | added | Jeff Harvey | Probably, as 708938760+240 =190373976+3*91171899+44330496+2*3376737+2572752+957125+61256+248 which has lower multiplicities than needed without adding 240. | |
| Feb 20, 2014 at 8:07 | comment | added | F. C. | Hum, maybe there is also a 240 in the coefficient 708938760, as it is the coefficient of $q^{16}$ ? | |
| Feb 20, 2014 at 4:03 | comment | added | Jeff Harvey | Not right off hand, but will take a look. | |
| Feb 19, 2014 at 23:56 | comment | added | Tito Piezas III | I'm sure Borcherds has a reason why it is defined that way. Do you know the functions that use $E_k$ for $k = 10,14$? | |
| Feb 19, 2014 at 23:54 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Added more detail.
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| Feb 19, 2014 at 23:03 | comment | added | Jeff Harvey | Aren't you tempted to get rid of the annoying factors of 240 by adding $120 \vartheta(0,\tau)$? | |
| Feb 19, 2014 at 22:44 | comment | added | Jeff Harvey | 708938760=190373976+3*111321000+2*91171899+1707264+6*85995+30628+4123+1. | |
| Feb 19, 2014 at 21:16 | comment | added | Tito Piezas III | Based on the original moonshine, I think sums of small multiples of the terms are allowed. | |
| Feb 19, 2014 at 21:13 | comment | added | F. C. | The next coefficient 708938760 seems to be harder to decompose.. | |
| Feb 19, 2014 at 20:52 | comment | added | Tito Piezas III | Ah, missed that! I added more terms to include 91171899. Thanks. | |
| Feb 19, 2014 at 20:51 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Added more terms.
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| Feb 19, 2014 at 20:46 | comment | added | F. C. | One also has 91951146 = 779247 + 91171899. | |
| Feb 19, 2014 at 20:42 | history | edited | Tito Piezas III | CC BY-SA 3.0 |
Made concise.
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| Feb 19, 2014 at 19:52 | history | asked | Tito Piezas III | CC BY-SA 3.0 |