Bases for modules of differential operators

Abstract

It is well-known that the derivation modules of Coxeter arrangements are free. Holm began to study the freeness of modules of differential operators on hyperplane arrangements. In this paper, we study the cases of the Coxter arrangements of type A, B and D. In this case, we prove that the modules of differential operators of order 2 are free. We give examples of all the 3-dimensional classical Coxeter arrangements. Two keys for the proof are ``Cauchy–Sylvester's theorem on compound determinants'' and ``Saito–Holm's criterion''.

Keywords

• coxeter arrangement
• cauchy―sylvester's compound determinants
• schur functions
• [info.info-dm] computer science [cs]/discrete mathematics [cs.dm]