Weights, Weyl-Equivariant Maps and a Rank Conjecture

Abstract

In this note, given a pair $(\mathfrak{g},\lambda ),$ where $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda \in {\mathfrak{h}}^{*}$ is a dominant integral weight of $\mathfrak{g},$ where $\mathfrak{h}\subset \mathfrak{g}$ is the real span of the coroots inside a fixed Cartan subalgebra, we associate an SU(2) and Weyl equivariant smooth map $f:X\to {(P^m(\mathbb{C}))}^n,$ where $X\subset \mathfrak{h}\otimes {\mathbb{R}}^3$ is the configuration space of regular triples in $\mathfrak{h},$ and m, n depend on the initial data $(\mathfrak{g},\lambda ).$ We conjecture that, for any $\mathbf{x}\in X,$ the rank of $f(\mathbf{x})$ is at least the rank of a collinear configuration in X (collinear when viewed as an ordered r-tuple of points in ${\mathbb{R}}^3,$ with r being the rank of $\mathfrak{g}$). A stronger conjecture is also made using the singular values of a matrix representing $f(\mathbf{x}).$ This work is a generalization of the Atiyah-Sutcliffe problem to a Lie-theoretic setting.

Keywords:

• Atiyah-Sutcliffe problem
• configurations of points
• root systems
• weights
• Weyl-equivariant maps

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.