# Modular Equalities for Complex Reflection Arrangements

### Anca Daniela Mačinic

Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania### Ştefan Papadima

Simion Stoilow Institute of Mathematics, P.O. Box 1-764, RO-014700 Bucharest, Romania### Clement Radu Popescu

Simion Stoilow Institute of Mathematics, unit no. 4, P.O. Box 1-764, RO-014700 Bucharest, Romania

## Abstract

We compute the combinatorial Aomoto-Betti numbers $β_{p}(A)$ of a complex reflection arrangement. When $A$ has rank at least 3, we find that $β_{p}(A)≤2$, for all primes $p$. Moreover, $β_{p}(A)=0$ if $p>3$, and $β_{2}(A)=0$ if and only if $A$ is the Hesse arrangement. We deduce that the multiplicity $e_{d}(A)$ of an order $d$ eigenvalue of the monodromy action on the first rational homology of the Milnor fiber is equal to the corresponding Aomoto-Betti number, when $d$ is prime. We give a uniform combinatorial characterization of the property $e_{d}(A)=0$, for $2≤d≤4$. We completely describe the monodromy action for full monomial arrangements of rank 3 and 4. We relate $e_{d}(A)$ and $β_{p}(A)$ to multinets, on an arbitrary arrangement.

## Cite this article

Anca Daniela Mačinic, Ştefan Papadima, Clement Radu Popescu, Modular Equalities for Complex Reflection Arrangements. Doc. Math. 22 (2017), pp. 135–150

DOI 10.4171/DM/561