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Configurations of Points and the Symplectic Berry–Robbins Problem
Symmetry, integrability and geometry: methods and applications, Tome 10 (2014) Cet article a éte moissonné depuis la source Math-Net.Ru

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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 $\operatorname{Sp}(n)$, instead of the Lie group $\operatorname{U}(n)$. Denote by $\mathfrak{h}$ a Cartan algebra of $\operatorname{Sp}(n)$, and $\Delta$ the union of the zero sets of the roots of $\operatorname{Sp}(n)$ tensored with $\mathbb{R}^3$, each being a map from $\mathfrak{h} \otimes \mathbb{R}^3 \to \mathbb{R}^3$. We wish to construct a map $(\mathfrak{h} \otimes \mathbb{R}^3) \backslash \Delta \to \operatorname{Sp}(n)/T^n$ which is equivariant under the action of the Weyl group $W_n$ of $\operatorname{Sp}(n)$ (the symplectic Berry–Robbins problem). Here, the target space is the flag manifold of $\operatorname{Sp}(n)$, 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 $n=2$.
Keywords: configurations of points; symplectic; Berry–Robbins problem; equivariant map; Atiyah–Sutcliffe problem. 
@article{SIGMA_2014_10_a111,
     author = {Joseph Malkoun},
     title = {Configurations of {Points} and the {Symplectic} {Berry{\textendash}Robbins} {Problem}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2014},
     volume = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a111/}
}
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Joseph Malkoun. Configurations of Points and the Symplectic Berry–Robbins Problem. Symmetry, integrability and geometry: methods and applications, Tome 10 (2014). http://geodesic.mathdoc.fr/item/SIGMA_2014_10_a111/

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