Configurations of Points and the Symplectic Berry-Robbins Problem

We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group $\mathrm{Sp}(n)$, instead of the Lie group $\mathrm{U}(n)$. Denote by $\mathfrak{h}$ a Cartan algebra of $\mathrm{Sp}(n)$, and $\mathrm{\Delta }$ the union of the zero sets of the roots of $\mathrm{Sp}(n)$ tensored with ${\mathbb{R}}^3$, each being a map from $\mathfrak{h} \otimes $\mathbb{R}$^3 \to $\mathbb{R}$^3$\mathrm{.}Wewishtoconstructamap$(\mathfrak{h} \otimes $\mathbb{R}$^3) \backslash \Delta \to \operatorname{Sp}(n)/T^n$ which is equivariant under the action of the Weyl group $W_n$ of $\mathrm{Sp}(n)$ (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of $\mathrm{Sp}(n)$, and $T^n$ is the diagonal $n$-torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for $n=2$.