2019/04/12 by Joseph Malkoun, Malkoun, Joseph
Mathematics · #11Cxx #15A18 #22E60 #33C52 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:11Cxx #msc:15A18 #msc:22E60 #msc:33C52
paper · pdf · doi:10.48550/arxiv.1904.06426
8 pages, 1 table of numerical values
arxiv created 2019/04/12 · arxiv updated 2019/04/16
In this note, given a pair (\mathfrakg, λ), where \mathfrakg is a complex semisimple Lie algebra and λ∈ \mathfrakh^* is a dominant integral weight of \mathfrakg, where \mathfrakh ⊂ \mathfrakg is the real span of the coroots inside a fixed Cartan subalgebra, we associate an SU(2) and Weyl equivariant smooth map f: X → (Pm(ℂ))n, where X ⊂ \mathfrakh ⊗ ℝ3 is the configuration space of regular triples in \mathfrakh, and m, n depend on the initial data (\mathfrakg, λ). We conjecture that, for any x ∈ X, the rank of f(x) is at least the rank of a collinear configuration in X (collinear when viewed as an ordered r-tuple of points in ℝ3, with r being the rank of \mathfrakg). A stronger conjecture is also made using the singular values of a matrix representing f(x). This work is a generalization of the Atiyah-Sutcliffe problem to a Lie-theoretic setting.