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Symmetry, Integrability and Geometry: Methods and Applications, 2014, Volume 10, 112, 6 pp.
DOI: https://doi.org/10.3842/SIGMA.2014.112
(Mi sigma977)
 

This article is cited in 2 scientific papers (total in 2 papers)

Configurations of Points and the Symplectic Berry–Robbins Problem

Joseph Malkoun

Department of Mathematics and Statistics, Notre Dame University-Louaize, Lebanon
Full-text PDF (251 kB) Citations (2)
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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.
Received: August 23, 2014; in final form December 17, 2014; Published online December 19, 2014
Bibliographic databases:
Document Type: Article
MSC: 51F99; 17B22
Language: English
Citation: Joseph Malkoun, “Configurations of Points and the Symplectic Berry–Robbins Problem”, SIGMA, 10 (2014), 112, 6 pp.
Citation in format AMSBIB
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\by Joseph~Malkoun
\paper Configurations of Points and the Symplectic Berry--Robbins Problem
\jour SIGMA
\yr 2014
\vol 10
\papernumber 112
\totalpages 6
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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