We establish an optimal $$L^p$$ -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions $$n\ge 5$$ : $$\begin{aligned} \Delta ^2 u=\Delta (D\cdot \nabla u)+\text {div}(E\cdot u) +(\Delta \Omega +G)\cdot u +f \qquad \ \mathrm{{in}}\ B^n, \end{aligned}$$ where $$\Omega \in W^{1,2}(B^n, so_m)$$ is and $$f\in L^p(B^n)$$ , $$D, ...

Lecture notes by Matthew Dyer for lectures at the workshop “Coxeter groups and convex geometry.” 1. Root labelled Bruhat graph 1.1. Bruhat order. Let (W,S) be a Coxeter group. Bruhat order is the partial order defined by the following proposition; see [3] and [13] for more details. Proposition. There is a partial order ≤ on W with the following properties. Let w ∈ W and w = s1 . . . sn, n = l0(...