I would like to define standard Gram matrices, and use them to help me understand the symmetries of lattices.


I define "standard Gram matrix" as the Gram matrix g that minimizes the deviation from Toeplitz and that satisfies abs(inv(g)) = abs(g).


I have obtained Gram matrices for D4, E8, A15+ with small deviations from Toeplitz that satisfy abs(inv(g)) = abs(g) but I cannot be sure that I have obtained the global optimum. My method for solving the optimization problem specified by the definition of "standard Gram matrix" is ad hoc and I cannot be sure that it has found the global optimum.


QUESTIONS:


-- How do I uniquely determine a standard Gram matrix for a lattice? Are there any other definitions of "standard Gram matrix" for lattices? Do you have any literature references in which "standard Gram matrix" is defined for any reason?


I seem to be able to uniquely define standard Gram matrices for D4, E8, A15+ for instance.


But I may have missed the optimum for E8 and A15+.


**This is my ad-hoc method for determining the standard Gram matrix according to my definition:**


I find a Gram matrix g for which abs(inv(g)) - abs(g) is zero. g seems to be unique up to permutations and sign flips, but I am not sure. I choose g to have the most positive signs. inv(mat) is the matrix inverse of mat, and abs(mat) is the absolute values of the matrix elements.


Furthermore I permute the basis vectors to make the Gram matrix maximally Toeplitz.


That seems to uniquely define the standard Gram matrices for D4 and E8.


("maximally" refers to the L1 norm of the matrix elements, sum of absolute values of deviations of matrix elements)


For instance, for the 24-cell D4 = D4\* lattice I obtain a Toeplitz Gram matrix


$$ \mathrm{gd4} = \sqrt{\frac{1}{2}} \left( \begin{array}{cccc} 
2 & 1 & 0 &-1 \\ 
1 & 2 & 1 & 0 \\ 
0 & 1 & 2 & 1 \\ 
-1& 0 & 1 & 2 \\ 
\end{array} \right)$$


The inverse is


$$ \mathrm{inv}(\mathrm{gd4}) = \sqrt{\frac{1}{2}} \left( \begin{array}{cccc} 
2 &-1 & 0 & 1 \\ 
-1& 2 &-1 & 0 \\ 
0 &-1 & 2 &-1 \\ 
1 & 0 &-1 & 2 \\ 
\end{array} \right) $$


likewise E8 has gram matrix ge8, not quite Toepliz.


$$ \mathrm{ge8 =} \begin{array}{ccccccccc} 
 && 2 && 0 && 1 && 0 && 0 && 0 && -1 && 1 \\
 && 0 && 2 && 1 && 1 && 0 && 0 && 0 && -1 \\
 && 1 && 1 && 2 && 0 && 1 && 0 && 0 && 0 \\
 && 0 && 1 && 0 && 2 && -1 && 1 && 0 && 0 \\
 && 0 && 0 && 1 && -1 && 2 && 0 && 1 && 0 \\
 && 0 && 0 && 0 && 1 && 0 && 2 && 1 && 1 \\
 && -1 && 0 && 0 && 0 && 1 && 1 && 2 && 0 \\
 && 1 && -1 && 0 && 0 && 0 && 1 && 0 && 2 \\
\end{array} $$


Each basis vector is perpendicular (90 degrees) to four other basis vectors, and makes angles of 60, 60, 120 degrees with the other three basis vectors.


inverse


$$ \mathrm{inv(ge8) =} \begin{array}{ccccccccc} 
 && 2 && 0 && -1 && 0 && 0 && 0 && 1 && -1 \\
 && 0 && 2 && -1 && -1 && 0 && 0 && 0 && 1 \\
 && -1 && -1 && 2 && 0 && -1 && 0 && 0 && 0 \\
 && 0 && -1 && 0 && 2 && 1 && -1 && 0 && 0 \\
 && 0 && 0 && -1 && 1 && 2 && 0 && -1 && 0 \\
 && 0 && 0 && 0 && -1 && 0 && 2 && -1 && -1 \\
 && 1 && 0 && 0 && 0 && -1 && -1 && 2 && 0 \\
 && -1 && 1 && 0 && 0 && 0 && -1 && 0 && 2 \\
\end{array} $$


The "Dynkin diagrams" are a square for D4 and a cube for E8 as shown in the figure here: [Dynkin diagrams for D4 and E8](https://i.stack.imgur.com/xRSno.png)


In contrast the usual Dynkin diagram for D4 is a triangle: <https://commons.wikimedia.org/wiki/File:Dynkin_diagram_D4.png>


I wonder whether I have found the maximally Toeplitz Gram matrix for E8 and A15+.


For A15+ the permutations are irrelevant. In this case I obtain a Gram matrix ga15plus satisfying


abs(ga15plus) = abs(inv(ga15plus))


$$ \mathrm{ga15plus \ =} \begin{array}{ccccccccccccccccc} 
 && 2 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 2 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 2 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 2 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 2 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 2 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 2 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 4 && 4 && 4 && 4 && 4 && 4 && 4 && 15 && 4 && 4 && 4 && 4 && 4 && 4 && 4 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 2 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 2 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 2 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 2 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 2 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 2 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && 4 && 1 && 1 && 1 && 1 && 1 && 1 && 2 \\
\end{array} $$


$$ \mathrm{inv(ga15plus) \ =} \begin{array}{ccccccccccccccccc} 
 && 2 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 2 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 2 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 2 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 2 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 2 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 2 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && -4 && -4 && -4 && -4 && -4 && -4 && -4 && 15 && -4 && -4 && -4 && -4 && -4 && -4 && -4 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 2 && 1 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 2 && 1 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 2 && 1 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 2 && 1 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 2 && 1 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 2 && 1 \\
 && 1 && 1 && 1 && 1 && 1 && 1 && 1 && -4 && 1 && 1 && 1 && 1 && 1 && 1 && 2 \\
\end{array} $$


With this Gram matrix there is one unique basis vector, so for A15+, there are 15 equivalent permutations, 15 equivalent standard gram matrices according to my definition.


But perhaps a different ga15plus also satisfies the inverse equality and can be made more Toeplitz.


In summary with my draft definition of "standard Gram matrix",


I seem to obtain unique standard Gram matrices for D4 and E8 and 15 equivalent standard Gram matrices for A15+. But I am not sure that I have obtained the most Toeplitz Gram matrices for E8 and A15+.


Again my questions are


-- How do I uniquely determine a standard Gram matrix for a lattice? Are there any other definitions of "standard Gram matrix" for lattices? ​Do you have any literature references in which "standard Gram matrix" is defined for any reason?



Standard Gram matrices for lattices

> 
> How do I uniquely determine a standard Gram matrix for a lattice? Are there any other definitions of "standard Gram matrix" for lattices? Do you have any literature references in which "standard Gram matrix" is defined for any reason?
> 
> 
> 


Yes, this has been done in [A Canonical Form for Positive Definite Matrices](https://arxiv.org/pdf/2004.14022.pdf). In particular, for positive definite $A$ (Gram matrices of positive definite lattices are positive definite), they
first define $A, B$ to be *arithmetically equivalent* if $A = U^tBU$ for uniomdular $U$.


They then define a mapping


$$A\mapsto Can(A)$$
such that


1. $Can(A)$ is equivalent to $A$, and
2. For any unimodular $U$, $Can(U^tAU)=Can(A)$.


Their (broad) strategy is to reduce to the case of graphs, for which there are recent quasi-polynomial time algorithms. This somewhat limits the maximum dimension they can handle (computationally) --- it appears they go up to dimension 40 at the most.


Note that this problem is thought to be hard though --- it would imply a solution to the *Lattice Isomorphism Problem*.
Efficient algorithms for this have been open for quite a while (there has been some interest for 10+ years), and recently [two](https://eprint.iacr.org/2021/1332.pdf) [cryptosystems](https://eprint.iacr.org/2021/1548) have been proposed assuming (variants of) this problem are hard, so now one can concretely say that efficient algorithms would break cryptographic proposals (so if you find one, there is a more obvious motivation to write it up).


