Abstract
In this note, given a pair where
is a complex semisimple Lie algebra and
is a dominant integral weight of
where
is the real span of the coroots inside a fixed Cartan subalgebra, we associate an SU(2) and Weyl equivariant smooth map
where
is the configuration space of regular triples in
and m, n depend on the initial data
We conjecture that, for any
the rank of
is at least the rank of a collinear configuration in X (collinear when viewed as an ordered r-tuple of points in
with r being the rank of
). A stronger conjecture is also made using the singular values of a matrix representing
This work is a generalization of the Atiyah-Sutcliffe problem to a Lie-theoretic setting.
Acknowledgements
I dedicate this work to Sir Michael Atiyah who came up with the original problem, as well as the question which motivated this work. The author thanks Ben Webster and James Humphreys for their comments on the Mathematics StackExchange website and by email. Any possible mistake in this work is however only the author’s responsibility. I would like to also thank the anonymous reviewer, whose comments made me add more details here and there, resulting in a clearer manuscript.
Declaration of Interest
No potential conflict of interest was reported by the author.