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JOURNALS // Symmetry, Integrability and Geometry: Methods and Applications // Archive

SIGMA, 2014 Volume 10, 112, 6 pp. (Mi sigma977)

This article is cited in 2 papers

Configurations of Points and the Symplectic Berry–Robbins Problem

Joseph Malkoun

Department of Mathematics and Statistics, Notre Dame University-Louaize, Lebanon

Abstract: 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.

MSC: 51F99; 17B22

Received: August 23, 2014; in final form December 17, 2014; Published online December 19, 2014

Language: English

DOI: 10.3842/SIGMA.2014.112



Bibliographic databases:
ArXiv: 1407.8291


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