Relation between different $E_8$ matrices

Question

There are several rank-8 square matrices known to be related to $E_8$:

1. Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix

$$M_1=\left [\begin{array}{rr} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 2 \end{array}\right ]$$

2. One choice of simple roots of $E_8$ is given by the rows of the following matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Simple_roots

$$M_2=\left [\begin{array}{rr} 1&-1&0&0&0&0&0&0 \\ 0&1&-1&0&0&0&0&0 \\ 0&0&1&-1&0&0&0&0 \\ 0&0&0&1&-1&0&0&0 \\ 0&0&0&0&1&-1&0&0 \\ 0&0&0&0&0&1&1&0 \\ -\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}&-\frac{1}{2}\\ 0&0&0&0&0&1&-1&0 \end{array}\right ]$$

3. $E_8$ lattice https://en.wikipedia.org/wiki/E8_lattice#Properties

$$M_3=\left[\begin{matrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 1/2 \\ 0 & 1 & -1 & 0 & 0 & 0 & 0 & 1/2 \\ 0 & 0 & 1 & -1 & 0 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 1 & -1 & 0 & 0 & 1/2 \\ 0 & 0 & 0 & 0 & 1 & -1 & 0 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 1 & -1 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1/2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 \end{matrix}\right]$$

I have also seen: $$M_4=\left [\begin{array}{rr} 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 2 & 1& 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 2 \end{array}\right ],$$ also $$M_5=\left [\begin{array}{rr} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1& 0 & 0 & 0 & 0 & 2 \end{array}\right ]$$

My question is that

• What are relations between different $E_8$ matrices? $M_1, M_2, M_3, M_4, M_5$ (Of course, they are related to $E_8$. They are all unimodular.) Could they be transformed to one from another, vice versa? (like $GL(8,\mathbf{Z})$?) What are the meanings of these transformations?

Answer 1

$M_1$, $M_4$ and $M_5$ are all the same quadratic form up to integer change of basis: To get from $M_1$ to $M_4$, conjugate by the diagonal matrix with digaonal entries $(1,-1,1,-1,1,-1,1,-1)$; to get from $M_1$ to $M_5$, conjugate by the permutation matrix $76543218$.

We have $M_2 M_2^T = M_1$, in other words, the rows of $M_2$ realize the quadratic form $M_1$, and the rows of $M_2$ are a basis for the $E_8$ lattice. (This is always the relationship between the simple roots and the Cartan matrix.)

$M_3$ is a matrix whose columns form a basis for the $E_8$ lattice. It is related to $M_2$ by $$M_3 = M_2^T \begin{bmatrix} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ \end{bmatrix}.$$ (This matrix has determinant $1$.) Thanks to Dave Benson in the comments for pointing this out. It looks like $M_3$ might be something like the Hermite normal form of $M_2^T$; it doesn't exactly match the Wikipedia definition of HNF, but there are many variants on Hermite normal form in the literature.

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As an overview: the columns of $M_2^T$ and $M_3$ are each bases of the same lattice. The matrices $M_1$, $M_4$ and $M_5$ are Gram matrices of various bases for that lattice -- $M_1$ is the Gram matrix in the columns of $M_2^T$, and the other two come from permuting and switching the signs of those basis vectors.

Answer 2

A euclidean lattice can be defined by a generating basis, for example as a nonsingular $n \times n$ real matrix $F$ whose columns generate the lattice which consists of all vectors $Fx$, $x \in \mathbb{Z}^n$. The norm (squared) of the lattice vectors is then given by the quadratic form $f(x)=||Fx||^2=x^TAx$, where $A=F^TF$ is the Gram matrix. If $F'$ is a different basis, then $F'=FG$ for some $G \in GL(n, \mathbb{Z})$, so the Gram matrix transforms as $A'={F'}^{T} F'=G^TAG$. Passing from $F$ to $A$ is just $A=F^TF$ but a rotated (and isometric) $F_1=KF, K \in O(n)$ gives the same $A$.

Conversely given a symmetric positive definite $A$, one can always factor $A=F^TF$ for example by Cholesky decomposition to get a basis $F$. This is just the method of completing squares. If one have two different such decomposition $A=F^TF=F_1^TF_1$, then $F_1$ is a rotation of $F$ and defines isometric lattice. So giving a basis $F$ or its quadratic form $A$ is equivalent.

So $M_2,M_3$ are basis vectors while $M_1,M_4,M_5$ are the quadratic forms. Since $E8$ is the only even unimodular lattice in dimension $8$,every rank 8 integral positive definite symmetric matrix with determinant $\pm 1$ and even on the diagonal will be its Gram matrix which one can verify to hold for $A=M_j,j=1,4,5$ and $A=M_k^TM_k,k=2,3$. The tedious part is to prove positive definiteness but one can compute the characteristic polynomial eg $\phi_{M_5}(x)=x^8-16x^7+105x^6-364x^5+714x^4-784^3+440x^2-96x+1,$ which has alternative sign so all eigenvalues are positive.

Finding such matrix is non-trivial as it is equivalent to proving existence of the even unimodular lattice. Try a 24D example which exist because of the Leech lattice. There is also a 72D example whose existence is open for decades and finally found by G. Nebe