2012/01/07 by James Damon, G. Bruce Pike, Damon, James +2
Mathematics · #11S90 (Secondary) #17B66 (Primary) 22E27 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #math.AG #math.RT #msc:11S90 #msc:17B66 #msc:22E27
paper · pdf · doi:10.48550/arxiv.1201.1577
50 pages. Many changes from v1 in response to a thorough review, mostly concentrated in sections 2, 3, and 4. To appear in Annales de l'Institut Fourier
openalex publication_date 2012/01/07 · arxiv created 2015/01/28 · arxiv updated 2015/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a method for obtaining new classes of free divisors from representations V of connected linear algebraic groups G where dim(G)=dim(V), with V having an open orbit. We give sufficient conditions that the complement of this open orbit, the "exceptional orbit variety", is a free divisor (or a slightly weaker free* divisor) for "block representations" of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated "representation vector fields" on V has block triangular form, with blocks satisfying certain nonsingularity conditions. For towers of Lie groups and representations this yields a tower of free divisors, successively obtained by adjoining varieties of singular matrices. This applies to solvable groups which give classical Cholesky-type factorization, and a modified form of it, on spaces of m × m symmetric, skew-symmetric or general matrices. For skew-symmetric matrices, it further extends to representations of nonlinear infinite dimensional solvable Lie algebras.