Timeline for Standard Gram matrices for lattices
Current License: CC BY-SA 4.0
28 events
| when toggle format | what | by | license | comment | |
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| Feb 11, 2022 at 6:11 | vote | accept | Dan Haxton | ||
| Feb 9, 2022 at 23:44 | answer | added | Mark Schultz-Wu | timeline score: 2 | |
| Jan 16, 2022 at 18:54 | comment | added | Dan Haxton | My ge8 and gd4 are determined up to sign flips & ordering by abs(g) = abs(inv(g)). If I reorder my ge8 and flip a sign then you can see how E8 has an (A2 - cross - D4) structure, $$\mathrm{ge8=}\begin{array}{cccccccc} 2& 1& 0&-1& 1& 0& 0& 0\\ 1& 2& 1& 0& 0& 1& 0& 0\\ 0& 1& 2& 1& 0& 0& 1& 0\\ -1& 0& 1& 2& 0& 0& 0& 1\\ 1& 0& 0& 0& 2&-1& 0& 1\\ 0& 1& 0& 0&-1& 2&-1& 0\\ 0& 0& 1& 0& 0&-1& 2&-1\\ 0& 0& 0& 1& 1& 0&-1& 2\\ \end{array}$$ This is $$\mathrm{ge8=}\begin{array}{cc} \sqrt{2}\mathrm{gd4}& I\\ I& \sqrt{2}\mathrm{inv(gd4)}\\ \end{array}$$ | |
| Jan 13, 2022 at 23:02 | comment | added | Dan Haxton | .. and perhaps "skew circulant" or something else is a principle for standardization better than "Toeplitz". Toeplitz includes skew circulant. I need to understand how and why skew circulant Gram matrices arise. Notice that my gram matrix for D4, gd4, is exactly skew circulant and for E8 ge8 is very nearly skew circulant | |
| Jan 13, 2022 at 18:01 | comment | added | Dan Haxton | .. the Toeplitz requirement is irrelevant for the diagram. The diagram is determined by abs(g) regardless of sign flips and permutations in g. The requirement abs(g) = abs(inv(g)) seems to determine the diagram for a qualifying self-dual lattice. If a self-dual lattice has abs(g) /= abs(inv(g)) then the diagram would seem to be undetermined because the diagram for the self-dual lattice (determined by abs(g)) and that of its dual (determined by abs(inv(g))) would almost certainly be different | |
| Jan 13, 2022 at 17:51 | comment | added | Dan Haxton | My method for optimization is ad hoc. It assumes that the g that solves abs(g) = abs(inv(g)) is unique up to sign flips & permutations. So first I find a solution of that integer optimization problem & then I perform an exhaustive search of sign flips & permutations to make the matrix most Toeplitz and most positive. I suspect that this optimization method is not sufficient because I suspect that the solution of abs(g) = abs(inv(g)) is actually not unique, but so far I have obtained unique gram matrices g for the cases D4 E8 A15+, using several different starting guesses. | |
| Jan 13, 2022 at 17:41 | comment | added | Dan Haxton | I want abs(g) = abs(inv(g)) to be true for a self-dual lattice like A2, D4, E8, A15+. I want to generalize this formula for matrices that are not self-dual like E7, E7*, probably by minimizing instead of zeroing the norm of abs(g) - abs(inv(g)) | |
| Jan 13, 2022 at 17:41 | comment | added | Dan Haxton | The Toeplitz part of my definition is mainly there to determine the permutation of basis vectors, to produce a unique result. The more important part of my definition is the requirement abs(g) = abs(inv(g)). I am very curious whether the g that solves abs(g) = abs(inv(g)) for the self-dual lattices is unique up to sign flips & permutations. | |
| Jan 13, 2022 at 16:42 | comment | added | Dan Haxton | The "deviation from Toeplitz" of a matrix mat that I chose is sum_ij abs(mat_ij - toep(mat)_ij). In other words it is a L1 deviation but I could have also chosen L2, the sum of squared deviations. toep(mat) is a function that returns the toeplitz part of matrix mat. the toeplitz part is gotten by averaging the diagonals, toep(mat)_ij = mean_(k-l = i-j)(mat_kl) | |
| Jan 13, 2022 at 15:34 | comment | added | Mark L. Stone | What is the exact criterion for "maximally Toeplitz" or "deviation from Toeplitz"? Presuming that criterion makes sense, that would result in a well-defined optimization problem, whose numerical solution could be sought. You need to separate (not conflate) the specification of the optimization problem from exact or approximate methods for solving it. | |
| Jan 13, 2022 at 10:15 | history | edited | YCor | CC BY-SA 4.0 |
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| Jan 13, 2022 at 9:57 | history | edited | YCor |
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| Jan 13, 2022 at 9:40 | comment | added | Dan Haxton | Or, the diagram for inv(gd4) s a tetrahedron in which the radius^2 of one of the nodes is twice the others (no decoration is required then; the diagram for a qualifying NxN gram matrix is generally a (N-1)-dim graph in which the radius of a node represents the length of the corresponding basis vector). Either way, the fact that the standard gd4 does not satisfy abs(g) = abs(inv(g)) seems unsatisfactory. I wonder to what degree the requirement abs(g) = abs(inv(g)) reduces the space of g, regardless of Toeplitz requirement I proposed, how this requirement classifies lattices by its degeneracy. | |
| Jan 13, 2022 at 8:56 | comment | added | Dan Haxton | The standard D4 gram matrix g, given by the tables of Nebe & Sloane for instance, corresponding to the triangular Coxeter-Dynkin diagram for D4 from Wiki commons.wikimedia.org/wiki/File:Dynkin_diagram_D4.png has an inverse inv(g) that does not satisfy abs(g) = abs(inv(g)) and so inv(g) has a different diagram which seems unsatisfactory. inv(g) has a tetrahedral Coxeter-Dynkin diagram, in which one node is decorated denoting that its basis vector has twice the squared length of the other three basis vectors. The cubic diagram for E8 seems maximally symmetric. | |
| Jan 13, 2022 at 8:18 | comment | added | Dan Haxton | Yes I mean "Coxeter diagram" not "Dynkin diagram" -- the diagram I am talking about, what I showed for D4 (square) and E8 (cube) and linked to Wiki for D4 (triangle) is defined for gram matrices for which the cosine of the angle between every pair of basis vectors is -0.5, 0, or 0.5 (angles 60, 90, 120). Each basis vector is a point, and lines connect pairs of points representing basis vectors at 60 or 120 degrees. Square has a higher symmetry order than triangle so I think that the square diagram for D4 given by my gram matrix is better than the triangle diagram on Wikipedia. | |
| Jan 13, 2022 at 7:23 | history | edited | Dan Haxton | CC BY-SA 4.0 |
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| Jan 13, 2022 at 7:17 | history | edited | Dan Haxton | CC BY-SA 4.0 |
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| Jan 13, 2022 at 7:11 | history | edited | Dan Haxton | CC BY-SA 4.0 |
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| Jan 13, 2022 at 7:04 | history | edited | Dan Haxton | CC BY-SA 4.0 |
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| Jan 13, 2022 at 6:59 | comment | added | Dan Haxton | I agree, "Perhaps the question you should be asking is how trying to determine the orthogonal symmetries of a lattice helps to make some non-arbitrary choices." Do you have any guidance regarding D4 and E8? | |
| Jan 13, 2022 at 6:56 | review | Close votes | |||
| Jan 28, 2022 at 3:07 | |||||
| Jan 13, 2022 at 6:54 | history | edited | Dan Haxton | CC BY-SA 4.0 |
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| Jan 13, 2022 at 6:48 | comment | added | Dan Haxton | The most compact definition is the gram matrix g that minimizes the deviation from Toeplitz and that satisfies abs(inv(g)) = abs(g) | |
| Jan 13, 2022 at 6:39 | comment | added | Vladimir Dotsenko | No, you are asking "what do you think of my definition", yet you do not give a precise definition. | |
| Jan 13, 2022 at 6:36 | comment | added | Dan Haxton | vladimir-dotsenko I am asking how to uniquely define the Gram matrix for a lattice. I know that there seems to be no precise definition of "standard" uniquely defined Gram matrix in the literature. That is why I am asking. Please forgive my confusion about the nomenclature. I tried to define the diagram I am describing. For E8 it is actually a cube not an octagon. The diagram places lines between points representing basis vectors at 60 or 120 degrees. | |
| Jan 13, 2022 at 6:27 | comment | added | Vladimir Dotsenko | This is a very confusing question. First, you are asking about definitions, yet there is no single precise definition. Second, Dynkin diagrams are those corresponding to the "finite" case, there is no A15+ : perhaps you mean Coxeter diagrams. Finally, those diagrams encode reflection groups, or more generally abstract Coxeter groups, in particular, they encode a presentation by generators and relations, and as such are not at all arbitrary. Perhaps the question you should be asking is how trying to determine the orthogonal symmetries of a lattice helps to make some non-arbitrary choices. | |
| S Jan 13, 2022 at 5:13 | review | First questions | |||
| Jan 13, 2022 at 8:46 | |||||
| S Jan 13, 2022 at 5:13 | history | asked | Dan Haxton | CC BY-SA 4.0 |