Integral expressions for derivations of multiarrangements

TL;DR

This paper introduces integral expressions for bases of certain multiarrangements, extending previous work to complex reflection groups and proposing a conjectural basis for the extended Catalan arrangement.

Contribution

It provides explicit integral formulas for bases of specific multiarrangements, including complex reflection groups and a conjectural basis for the extended Catalan arrangement.

Findings

Integral expressions for bases of three-line arrangements in dimension 2.

Explicit bases for free multiarrangements related to complex reflection groups.

A conjectural basis for the module of logarithmic vector fields of the extended Catalan arrangement.

Abstract

The construction of an explicit basis for a free multiarrangement is not easy in general. Inspired by the integral expressions for quasi-invariants of quantum Calogero-Moser systems, we present integral expressions for specific bases of certain multiarrangements. Our construction covers the cases of three lines in dimension $2$ (previously examined by Wakamiko) and free multiarrangements associated with complex reflection groups (Hoge, Mano, R\"ohrle, Stump). Furthermore, we propose a conjectural basis for the module of logarithmic vector fields of the extended Catalan arrangement of type $B_2$.