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

摘要

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