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

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

My question is that

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

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?

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

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

My question is that

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

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

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

My question is that

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