2013/01/31 by Carlos Florentino, C. Florentino, Sean Lawton +1
Mathematics · #Abelian group #Abelian variety #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic group #Algebraic number #Character (mathematics) #Classifying space #Cohomology #Combinatorics #Discrete mathematics #Elementary abelian group #Finitely generated group #Finitely-generated abelian group #Free abelian group #Group (periodic table) #Group theory #Homotopy and Cohomology in Algebraic Topology #Irreducibility #Mathematical analysis #Mathematics #Pure mathematics #Rank of an abelian group #Reductive group #math.AG #math.GT #math.RT #msc:14L17 #msc:14L24 #msc:14L30 #msc:14P25 #msc:22E46
paper · pdf · doi:10.1016/j.topol.2014.05.009
published as Topology and its Applications, Volume 173, 15 August 2014, Pages 32-58 · 33 pages; version 3: few small changes, one error corrected, one or two additional references; to appear in Topology and its Applications
arxiv created 2014/05/12 · openalex publication_date 2014/05/27 · crossref created 2014/05/27 · arxiv updated 2014/06/11 · crossref issued 2014/08/01 · crossref published 2014/08/01 · crossref published-print 2014/08/01 · crossref deposited 2025/09/28 · openalex created_date 2025/10/10 · crossref indexed 2026/08/01 · openalex updated_date 2026/08/05
Let G be a complex reductive algebraic group (not necessarily connected), let K be a maximal compact subgroup, and let A be a finitely generated Abelian group. We prove that the conjugation orbit space Hom(A,K)/K is a strong deformation retract of the GIT quotient space Hom(A,G)//G. As a corollary, we determine necessary and sufficient conditions for the character variety Hom(A,G)//G to be irreducible when G is connected and semisimple. For a general connected reductive G, analogous conditions are found to be sufficient for irreducibility, when A is free abelian.