Constructions of $SU(2)$ and Weyl equivariant maps for all classical groups

Abstract

If $G$ is a compact Lie group, $T$ a maximal torus in $G$ (with Lie algebras $\mathfrak{g}$ and $\mathfrak{t}$ respectively) and $W$ the corresponding Weyl group, then the Berry-Robbins problem for $G$, as formulated by Sir Michael Atiyah and Roger Bielawski, asks whether there exists a continuous $SU(2)\times W$ equivariant map from the space of regular Cartan triples (an open subset of $\mathfrak{t}\otimes {\mathbb{R}}^3$) to $G/T$, where $SU(2)$ acts via a regular Lie group homomorphism $SU(2)\to G$. This was settled positively by Atiyah and Bielawski, and their maps are even smooth, but they are not explicit. For $G=U(n)$, there exists another construction due to Sir Michael Atiyah and developed further with Paul Sutcliffe, which is explicit, but relies on a linear independence conjecture. The author had previously found a similar type of construction for $G=Sp(m)$, also relying on a linear independence conjecture. In this paper, similar constructions are done for $SO(2m+1)$ and $SO(2m)$, thus exhausting the list of classical groups.

16 pages

References in corpus (2)

• A proof of Atiyah's conjecture on configurations, of four points in Euclidean three-space
• On the Conjectures Regarding the 4-Point Atiyah Determinant