Timeline for Lattices containing $A_n$ and $D_n$
Current License: CC BY-SA 4.0
15 events
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| Feb 6, 2022 at 15:24 | answer | added | David E Speyer | timeline score: 1 | |
| Feb 6, 2022 at 14:37 | comment | added | David E Speyer | Note that, even with rescaling, the dimension must be of the form $k^2-1$. Let $A$ and $D$ be the Gram matrices of the two lattices. Let $S$ be the rational change of basis matrix which turns the simple $A$-basis into the simple $D$-basis. Then $SAS^T = D$ so $\det(A) \det(S)^2 = \det(D)$. We have $\det(A) = n+1$ and $\det(D) = 4$, so we deduce that $\tfrac{n+1}{4}$ is a rational square. Given this, it would be nice to see whether or not $15$ dimensions is achievable. | |
| Feb 6, 2022 at 0:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 8, 2021 at 23:06 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Sep 8, 2021 at 23:25 | comment | added | Marco Golla | @LSpice: integral overlattices of an integral lattice $L$ correspond to isotropic (? I think this is the correct word) subgroups of the dual group of $L^*/L$. | |
| Sep 8, 2021 at 22:41 | history | edited | YCor |
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| Sep 8, 2021 at 22:14 | comment | added | LSpice | @MarcoGolla, dual group of what? | |
| Sep 8, 2021 at 22:03 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Aug 9, 2021 at 20:27 | answer | added | FermaX | timeline score: 0 | |
| Aug 3, 2021 at 21:55 | history | edited | Daniel Sebald | CC BY-SA 4.0 |
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| Aug 3, 2021 at 21:52 | comment | added | Daniel Sebald | Rescaling is fine. | |
| Aug 3, 2021 at 20:21 | comment | added | Marco Golla | If you don't admit rescalings (i.e. if you only consider integral lattices), then overlattices of $A_n$ and $D_n$ are classified by subgroups of the dual group. In particular $D_n$ has at most three overlattices (one or two of which are diagonal). | |
| Aug 3, 2021 at 20:14 | comment | added | Roland Bacher | There is a problem with the Leech lattice: it contains no roots. I think you accept rescalings. | |
| Aug 3, 2021 at 20:11 | comment | added | Roland Bacher | Such lattices must be unimodular (of determinant 1) and of dimension $k^2-1$ (since $A_n$ has determinant $n+1$). | |
| Aug 3, 2021 at 20:06 | history | asked | Daniel Sebald | CC BY-SA 4.0 |