Timeline for Integral positive definite quadratic forms and graphs
Current License: CC BY-SA 4.0
13 events
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| Oct 5, 2021 at 4:58 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
http -> https (the question has been bumped anyway)
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| Sep 3, 2010 at 13:45 | history | edited | VA. | CC BY-SA 2.5 |
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| Sep 3, 2010 at 13:39 | history | edited | VA. | CC BY-SA 2.5 |
added 145 characters in body; edited tags; added 10 characters in body; added 9 characters in body
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| Sep 3, 2010 at 9:01 | answer | added | Roland Bacher | timeline score: 3 | |
| Sep 2, 2010 at 15:25 | comment | added | VA. | @Victor: let $R$ be the set of all integral elements of $Z^n$ of square 2. Then $R$ is finite, spans $Z^n$ (because it contains $e_i$), and reflections in the elements $r\in R$ send $R$ to itself. It follows that $R$ is a reduced root system, a direct sum of $A_n,D_n,E_n$. So after changing a $\mathbb Z$-basis, we get one of the ADE graphs. In the original basis $e_i$, the graph may not be ADE. My question is: what is it? | |
| Sep 2, 2010 at 15:13 | comment | added | Victor Protsak | "Clearly, the standard basis vectors $e_i$ are roots (they have square 2), and generate the lattice $Z^n.$ By the standard result about root lattices, $Z^n$ is then a direct sum of the $A_n, D_n, E_n$ root lattices, and one can restrict to the case of a single direct summand." I cannot follow this at all: by definition, a root lattice is spanned by a root system, which is a finite set $R$ of vectors invariant under reflections in the hyperplanes orthogonal to the vectors from $R.$ So what is the root system associated with the matrix $A$? | |
| Sep 2, 2010 at 13:27 | history | edited | VA. | CC BY-SA 2.5 |
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| Sep 2, 2010 at 12:30 | comment | added | Gjergji Zaimi | Possibly a start could be the description of unit positive definite sincere quadratic forms given in "Sincere weakly positive unit quadratic forms" by M.V. Zeldich (Canadian Mathematical Society, Conference proceedings, Vol 14, 1993). Parts of it can be previewed here books.google.com/… | |
| Sep 2, 2010 at 9:23 | answer | added | Gjergji Zaimi | timeline score: 2 | |
| Sep 2, 2010 at 7:02 | comment | added | Gerry Myerson | @Robby, no, these are "Dynkin diagrams." See, e.g., en.wikipedia.org/wiki/Root_system | |
| Sep 2, 2010 at 6:13 | comment | added | Robby McKilliam | I know $K_n$ is the complete graph and $C_n$ is a cycle graph and I am assuming that $E_n$ is the empty graph. What are $A_n$ and $D_n$? | |
| Sep 2, 2010 at 1:01 | history | edited | VA. | CC BY-SA 2.5 |
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| Sep 1, 2010 at 14:18 | history | asked | VA. | CC BY-SA 2.5 |