Equivariant Chow functions and Wonderful Models in type A and B

Abstract

We prove a general inversion formula in the equivariant incidence algebras of geometric lattices that relates the equivariant Chow function with the equivariant super-reduced characteristic function. We apply this result to the case of minimal wonderful models for hyperplane arrangements associated to root systems, with the natural action of the corresponding Weyl group. Applying this formula to type $A$ (i.e. the partition lattices), we recover Getzler's compositional identity in type $A$. In type $B$ (i.e. the signed partition lattice), we recover Henderson's result, expressing the equivariant Chow generating series in terms of the same series in type $A$ and the equivariant characteristic generating series of type $B$. Moreover, we obtain a new compositional identity for the equivariant Chow generating series.

37 pages