A basis construction of the extended Catalan and Shi arrangements of the type $A_{2}$

TL;DR

This paper provides the first explicit basis construction for the logarithmic derivation modules of extended Catalan and Shi arrangements specifically for the type A2 root system, advancing understanding of their algebraic structure.

Contribution

It introduces the first explicit basis construction for these arrangements' derivation modules in the A2 case, filling a gap in the existing literature.

Findings

Explicit bases for A2 arrangements constructed

Advances understanding of arrangement freeness

Provides tools for further algebraic studies

Abstract

In [9], Terao proved the freeness of multi-Coxeter arrangements with constant multiplicities by giving an explicit construction of bases. Combining it with algebro-geometric method, Yoshinaga proved the freeness of the extended Catalan and Shi arrangements in [11]. However, there have been no explicit constructions of the bases for the logarithmic derivation modules of the extended Catalan and Shi arrangements. In this paper, we give the first explicit construction of them when the root system is of the type $A_2$.