On an exponent-preserving duality for Coxeter multiarrangements of type $B_2$

In the theory of hyperplane arrangements, the study of logarithmic derivation modules is one of the most important subjects. If the module is free over a graded ring, the multiset of degrees of its homogeneous basis is called exponents. We utilize some basic theory of linear algebra to study the exponents of multiarrangements. As an application, we show a new exponent-preserving duality for the Coxeter multiarrangements of type $B_2$, which was conjectured by Torsten Hoge.