implement the Coxeter polynomial

[fchapoton]

which is an interesting non-trivial polynomial invariant of posets, depending only on the derived category of modules.

This is the characteristic polynomial of the matrix

- self.lequal_matrix() * self.moebius_function_matrix().transpose()

See https://doc.sagemath.org/html/en/reference/combinat/sage/combinat/posets/posets.html#sage.combinat.posets.posets.FinitePoset.coxeter_polynomial

[fchapoton]

pseudo code, not tested:

InstallMethod(CoxeterPolynomial,
"for Poset",
[IsPoset],
function(X)
        zeta := OrderMatrix(X);
        mu := MobiusMatrix(X);
        cox := -zeta * TransposedMat(mu);
        return CharacteristicPolynomial(cox)
end);

[kpiterman]

You can implement it and run a few examples. If you have concrete computations that depend on this polynomial, you can add some examples for the documentation. Then we can see if we merge this with the main branch.

[fchapoton]

yes, but I have difficulties : CharacteristicPolynomial refuses this kind of matrix as input..