Hilbert–Poincaré series for spaces of commuting elements in Lie groups

2017/04/30 by Daniel A. Ramras, Mentor Stafa · 8 citations
Mathematics · #Adjoint representation #Advanced Algebra and Geometry #Central series #Equivalence (formal languages) #Geometric and Algebraic Topology #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Lie group #Nilpotent #Representation theory #Series (stratigraphy) #Simple Lie group #math.AT #math.RT #msc:20F55 #msc:22E99 #msc:55N10 #msc:57T10

paper · pdf · doi:10.1007/s00209-018-2122-1

published in Mathematische Zeitschrift 292(1-2), 591-610 (Springer Science+Business Media) · 20 pages, journal version

arxiv created 2018/07/25 · openalex publication_date 2018/07/31 · openalex created_date 2018/08/03 · arxiv updated 2019/08/02 · openalex updated_date 2026/08/05

Abstract

In this article we study the homology of spaces \rm Hom(ℤⁿ,G) of ordered pairwise commuting n-tuples in a Lie group G. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of G. By work of Bergeron and Silberman, our results also apply to \rm Hom(F_n/Γ_n^m,G), where the subgroups Γ_n^m are the terms in the descending central series of the free group F_n. Finally, we show that there is a stable equivalence between the space \rm Comm(G) studied by Cohen-Stafa and its nilpotent analogues.

Citations

• Finite Reflection Groups