Hyperplane arrangements and Vinberg's θ-groups

2025/09/04 by Ambrosio, Filippo, Santi, Andrea
#17B40 #17B70 (Primary) #20F55 (Secondary) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2509.04284

Abstract

Let \mathfrakg = \bigoplus_{i ∈ ℤ /m ℤ} \mathfrakg_i be a periodically graded semisimple complex Lie algebra. In this note, we give a uniform proof of the recent result by W. de Graaf and H. V. Lê that the hyperplane arrangement determined by the restrictions of the roots of \mathfrakg to a Cartan subspace \mathfrakc ⊂ \mathfrakg₁ coincides with the hyperplane arrangement of (complex) reflections of the little Weyl group of \mathfrakg = \bigoplus_{i ∈ ℤ /m ℤ} \mathfrakg_i.

Citations

• Families of curves in Vinberg representations
• Discriminants of Theta-Representations
• Classification of four-rebit states
• Classification of real trivectors in dimension nine
• ℤ/mℤ-graded Lie algebras and perverse sheaves, III: graded double affine Hecke algebra
• Z/m-graded Lie algebras and perverse sheaves, I
• Regular subalgebras and nilpotent orbits of real graded Lie algebras

Related

• On Invarinat Theory of θ-groups
• Vinberg's θ-groups in positive characteristic and Kostant-Weierstrass slices
• The intrinsic hyperplane arrangement in an arbitrary irreducible representation of the symmetric group
• Hyperplane arrangements associated to symplectic quotient singularities
• Orbit configuration spaces associated to discrete subgroups of PSL(2,R)