What is the significance of the eigenvalues and eigenvectors of the exceptional simple Lie group root lattice to the Lie group or other mathematics branches?
For example,
E6, we have $$ \left( \begin{array}{cccccc} 2 & -1 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 2 \\ \end{array} \right) $$ with eigenvalues: $${3.93185, 3., 2.51764, 1.48236, 1., 0.0681483}.$$
E7, we have $$ \left( \begin{array}{ccccccc} 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 2 \\ \end{array} \right)$$ with eigenvalues: $$ {3.96962, 3.28558, 2.68404, 2., 1.31596, 0.714425, 0.0303845} $$
E8, we have $$ \left( \begin{array}{cccccccc} 2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 2 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 2 & -1 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 2 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 2 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 2 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 & 2 & 0 \\ 0 & 0 & -1 & 0 & 0 & 0 & 0 & 2 \\ \end{array} \right) $$ with eigenvalues: $$ {3.98904, 3.48629, 2.81347, 2.41582, 1.58418, 1.18653, 0.51371, 0.0109562} $$ Of course we know eigenvalues can be used to built invariant such as determinant of the matrix. But here I ask the significance of the eigenvalues and eigenvectors of the exceptional simple Lie group or other mathematics branches?
For example E8 lattice can be used for the intersection form of E8 manifolds.
