On the Milnor monodromy of the exceptional reflection arrangement of type $G_{31}$

TL;DR

This paper determines the monodromy action on the first cohomology of the Milnor fiber for the exceptional complex reflection group G_{31}, completing the classification for all irreducible complex reflection groups.

Contribution

It provides the first detailed analysis of the monodromy for G_{31}, using spectral sequences and computational methods, filling a gap in the understanding of such arrangements.

Findings

Monodromy action on H^1(F,C) for G_{31} is explicitly computed.

The work completes the classification of monodromy operators for all irreducible complex reflection groups.

Combines theoretical results with computer-aided calculations.

Abstract

Combining recent results by A. Macinic, S. Papadima and R. Popescu with a spectral sequence and computer aided computations, we determine the monodromy action on $H^1(F,\mathbb{C})$, where $F$ denotes the Milnor fiber of the hyperplane arrangement associated to the exceptional irreducible complex reflection group $G_{31}$. This completes the description given by the first author of such monodromy operators for all the other irreducible complex reflection groups.