Hilbert-Poincaré series for spaces of commuting elements in Lie groups. (English) Zbl 1416.22023
Let \(G\) be a compact and connected Lie group \(G\) and \(\pi\) a discrete group \(\pi\) generated by \(n\) elements. The authors consider the rational homology of the space of group homomorphisms \(\operatorname{Hom}(\pi, G)\subseteq G^n\), endowed with the subspace topology from \(G^n\) and give an explicit formula for the Poincaré series of \(\operatorname{Hom}(\pi, G)_1\), the connected component of the trivial representation, when \(\pi\) is free abelian or nilpotent.
The formula given for the Poincaré series of the identity component \(\operatorname{Hom}(\mathbb{Z}^n, G)_1\) is based on the works [T. J. Baird, Algebr. Geom. Topol. 7, 737–754 (2007; Zbl 1163.57026)] and [F. R. Cohen and M. Stafa, Math. Proc. Camb. Philos. Soc. 161, No. 3, 381–407 (2016; Zbl 1371.55012)].
The formula given for the Poincaré series of the identity component \(\operatorname{Hom}(\mathbb{Z}^n, G)_1\) is based on the works [T. J. Baird, Algebr. Geom. Topol. 7, 737–754 (2007; Zbl 1163.57026)] and [F. R. Cohen and M. Stafa, Math. Proc. Camb. Philos. Soc. 161, No. 3, 381–407 (2016; Zbl 1371.55012)].
Reviewer: Osman Mucuk (Kayseri)
MSC:
| 22E99 | Lie groups |
| 55N10 | Singular homology and cohomology theory |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 57T10 | Homology and cohomology of Lie groups |
Keywords:
representation space; Hilbert-Poincaré series; characteristic degree; finite reflection groupReferences:
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