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.