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

In this article we study the homology of spaces Hom ( Z n , G ) of ordered pairwise commuting n -tuples in a Lie group G . We give an explicit formula for the Poincaré 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...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 292; no. 1-2; pp. 591 - 610
Main Authors Ramras, Daniel A., Stafa, Mentor
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2019
Springer Nature B.V
Subjects
Homology
Lie groups
Mathematics
Mathematics and Statistics
Subgroups
Finite reflection group
57T10
Representation space
55N10
Characteristic degree
Hilbert–Poincaré series
Primary 22E99
Secondary 20F55
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Summary:In this article we study the homology of spaces Hom ( Z n , G ) of ordered pairwise commuting n -tuples in a Lie group G . We give an explicit formula for the Poincaré 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 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 Comm ( G ) studied by Cohen–Stafa and its nilpotent analogues.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-018-2122-1