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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.

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