Title: The adjoint braid arrangement as a free Lie algebra in species via the Steinmann relations
Abstract: We describe a discrete derivative of piecewise constant functions on the adjoint braid arrangement with respect to binary layered trees. We show that this derivative is the cobracket of a Lie coalgebra internal to the category of linear species precisely when one restricts to functions which satisfy the so-called Steinmann relations. We prove that this restriction coincides with the span of characteristic functions of permutohedral cones, and that the kernel of the cobracket is spanned by characteristic functions of complete fans over systems of affine roots. This is joint work with Zhengwei Liu and Adrian Ocneanu.