On the extended Catalan arrangement of type 𝐵_{ℓ}

The extended Catalan arrangement is a class of finite truncation of an affine Weyl arrangement for a root system. Edelman and Reiner conjectured that the coning of the extended Catalan arrangement is free, and Yoshinaga gave an affirmative answer to this conjecture for all root systems. However, the explicit descriptions of their logarithmic vector field are not well known for a long time. In this article, we provide an explicit formula for the generators of the logarithmic vector field of the coning of the extended Catalan arrangement of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Subscript script l"> <mml:semantics> <mml:msub> <mml:mi>B</mml:mi> <mml:mi> ℓ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">B_\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular, we settle the conjecture of Feigin, Wang and Yoshinaga affirmatively.