  Trigonometric polynomials are usually defined on the lattice of integers.We
consider the larger class of weight and root lattices with crystallographic
symmetry.This article gives a new approach to minimize trigonometric
polynomials, which are invariant under the associated reflection group.The
invariance assumption allows us to rewrite the objective function in terms of
generalized Chebyshev polynomials. The new objective function is defined on a
compact basic semi-algebraic set, so that we can benefit from the rich theory
of polynomial optimization.We present an algorithm to compute the minimum:
Based on the Hol-Scherer Positivstellensatz, we impose matrix-sums of squares
conditions on the objective function in the Chebyshev basis.The degree of the
sums of squares is weighted, defined by the root system. Increasing the degree
yields a converging Lasserre-type hierarchy of lower bounds.This builds a
bridge between trigonometric and polynomial optimization, allowing us to
compare with existing techniques.The chromatic number of a set avoiding graph
in the Euclidean space is defined through an optimal coloring.It can be
computed via a spectral bound by minimizing a trigonometric polynomial. If the
to be avoided set has crystallographic symmetry, our method has a natural
application.Specifically, we compute spectral bounds for the first time for
boundaries of symmetric polytopes.For several cases, the problem has such a
simplified form that we can give analytical proofs for sharp spectral bounds.In
other cases, we certify the sharpness numerically.
