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Which linear combinations of simple roots are roots
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I guess the following is not exactly what you are looking for but is somewhat revealing. A former colleague of mine tackled these type of questions in his PhD thesis. He describes something that he ...

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Lattice question
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I think indeed the claim is true under some "weak" assumptions on the fundamental unit. For instance let $\langle u_i, x \rangle = b_i$ be the equations of the codimension $1$ faces of the fundamental ...

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