Modules of differential operators of order 2 on Coxeter arrangements

TL;DR

This paper proves that modules of differential operators of order 2 on classical Coxeter arrangements are free by constructing explicit bases using determinant theorems and Schur polynomials, extending results across types A, B, and D.

Contribution

It provides explicit bases for modules of differential operators of order 2 on classical Coxeter arrangements, utilizing determinant theorems and Schur polynomials, and extends the freeness results to types A, B, and D.

Findings

Modules of differential operators of order 2 are free on classical Coxeter arrangements.

Explicit bases are constructed using Cauchy-Sylvester's theorem and Schur polynomials.

The approach applies uniformly across types A, B, and D.

Abstract

We prove that the modules of differential operators of order 2 on the classical Coxeter arrangements are free by exhibiting bases. For this purpose, we use Cauchy-Sylvester's theorem on compound determinants and Saito-Holm's criterion. In the case type $A$, we apply Cauchy-Sylvester's theorem on compound determinants to Vandermond determinant. By using the Schur polynomials, we define operators which form a part of a basis of modules of differential operators on the classical Coxeter arrangements of type $A$. In the cases of type $B$ and type $D$, the proofs go similarly to the case of type $A$ with some adjustments of operators and determinants.