Invariant differential derivations for reflection groups in positive characteristic

Abstract

Much of the captivating numerology surrounding finite reflection groups stems from Solomon's celebrated 1963 theorem describing invariant differential forms. Invariant differential derivations also exhibit fascinating numerology over the complex numbers linked to rational Catalan combinatorics. We explore the analogous theory over arbitrary fields, in particular, when the characteristic of the underlying field divides the order of the acting reflection group and the conclusion of Solomon's Theorem may fail. Using results of Broer and Chuai, we give a Saito criterion (Jacobian criterion) for finding a basis of differential derivations invariant under a finite group that distinguishes certain cases over fields of characteristic 2. We show that the reflecting hyperplanes lie in a single orbit and demonstrate a duality of exponents and coexponents when the transvection root spaces of a reflection group are maximal. A set of basic derivations are used to construct a basis of invariant differential derivations with a twisted wedging in this case. We obtain explicit bases for the special linear groups $\text{SL}(n,q)$ and general linear groups $\text{GL}(n,q)$, and all groups in between.