direction.

The [*Milnor fiber*]{} of a degree $n$ homogeneous polynomial $f\in {\mathbb{C}}[x_1, \dots, x_l]$ is an important object in singularity theory, introduced by Milnor in [@Mi]. It is defined by $F_f := f^{-1}(1)$, and comes endowed with a natural automorphism, induced by multiplication in ${\mathbb{C}}^l$ by a primitive $n$-root of unity. The induced action on $H_*(F_f, {\mathbb{Q}})$ is called the [*algebraic monodromy*]{}. Its study is a very active area of research in singularity theory. In arrangement theory, the Milnor fiber of ${{\mathcal A}}$, $F_{{{\mathcal A}}}$, is associated to the defining polynomial of ${{\mathcal A}}$, $Q({{\mathcal A}}):= \Pi_{H\in {{\mathcal A}}} \alpha_H$.

A long-standing open problem in this context is the combinatorial determination of the algebraic monodromy action on $H_1(F_{{{\mathcal A}}}, {\mathbb{Q}})$. We address this question in Section \[sect:Milnor\]. Following [@MP], we present an affirmative answer, in terms of combinatorial objects defined in positive characteristic, for subarrangements of type $B$ Coxeter arrangements, parametrized by [*graphs*]{}. We also emphasize a general relation between algebraic monodromy and twisted cohomology, which is key to the proof.

Lattice and complement of an arrangement {#sect:lattice}
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Assume ${{\mathcal A}}\subset V,\;V$ a complex vector space of dimension $l$, is an arrangement of hyperplanes. Choosing a system of coordinates 