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 $(\\mathfrak{g}, \\lambda)$. \nWe 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 $f(\\mathbf{x})$. \nThis work is a generalization of the Atiyah-Sutcliffe problem to a Lie-theoretic setting.