Hilbert–Poincaré series for spaces of commuting elements in Lie groups
Abstract In this article we study the homology of spaces $$\mathrm{Hom}({\mathbb Z}^n,G)$$ 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, ou...
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| Autor*in: |
Ramras, Daniel A. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2018 |
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| Anmerkung: |
© Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
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| Übergeordnetes Werk: |
Enthalten in: Mathematische Zeitschrift - Springer Berlin Heidelberg, 1918, 292(2018), 1-2 vom: 31. Juli, Seite 591-610 |
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| Übergeordnetes Werk: |
volume:292 ; year:2018 ; number:1-2 ; day:31 ; month:07 ; pages:591-610 |
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10.1007/s00209-018-2122-1 |
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| 520 | |a Abstract In this article we study the homology of spaces $$\mathrm{Hom}({\mathbb Z}^n,G)$$ 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 $$\mathrm{Hom}(F_n/\Gamma _n^m,G)$$, where the subgroups $$\Gamma _n^m$$ 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 $$\mathrm{Comm}(G)$$ studied by Cohen–Stafa and its nilpotent analogues. | ||
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10.1007/s00209-018-2122-1 doi (DE-627)OLC2039747893 (DE-He213)s00209-018-2122-1-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Ramras, Daniel A. verfasserin aut Hilbert–Poincaré series for spaces of commuting elements in Lie groups 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In this article we study the homology of spaces $$\mathrm{Hom}({\mathbb Z}^n,G)$$ 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 $$\mathrm{Hom}(F_n/\Gamma _n^m,G)$$, where the subgroups $$\Gamma _n^m$$ 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 $$\mathrm{Comm}(G)$$ studied by Cohen–Stafa and its nilpotent analogues. Representation space Hilbert–Poincaré series Characteristic degree Finite reflection group Stafa, Mentor aut Enthalten in Mathematische Zeitschrift Springer Berlin Heidelberg, 1918 292(2018), 1-2 vom: 31. Juli, Seite 591-610 (DE-627)129474193 (DE-600)203014-7 (DE-576)014852047 0025-5874 nnns volume:292 year:2018 number:1-2 day:31 month:07 pages:591-610 https://doi.org/10.1007/s00209-018-2122-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4320 AR 292 2018 1-2 31 07 591-610 |
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10.1007/s00209-018-2122-1 doi (DE-627)OLC2039747893 (DE-He213)s00209-018-2122-1-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Ramras, Daniel A. verfasserin aut Hilbert–Poincaré series for spaces of commuting elements in Lie groups 2018 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag GmbH Germany, part of Springer Nature 2018 Abstract In this article we study the homology of spaces $$\mathrm{Hom}({\mathbb Z}^n,G)$$ 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 $$\mathrm{Hom}(F_n/\Gamma _n^m,G)$$, where the subgroups $$\Gamma _n^m$$ 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 $$\mathrm{Comm}(G)$$ studied by Cohen–Stafa and its nilpotent analogues. Representation space Hilbert–Poincaré series Characteristic degree Finite reflection group Stafa, Mentor aut Enthalten in Mathematische Zeitschrift Springer Berlin Heidelberg, 1918 292(2018), 1-2 vom: 31. Juli, Seite 591-610 (DE-627)129474193 (DE-600)203014-7 (DE-576)014852047 0025-5874 nnns volume:292 year:2018 number:1-2 day:31 month:07 pages:591-610 https://doi.org/10.1007/s00209-018-2122-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_70 GBV_ILN_2007 GBV_ILN_2018 GBV_ILN_2030 GBV_ILN_4027 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4306 GBV_ILN_4320 AR 292 2018 1-2 31 07 591-610 |
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Abstract In this article we study the homology of spaces $$\mathrm{Hom}({\mathbb Z}^n,G)$$ 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 $$\mathrm{Hom}(F_n/\Gamma _n^m,G)$$, where the subgroups $$\Gamma _n^m$$ 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 $$\mathrm{Comm}(G)$$ studied by Cohen–Stafa and its nilpotent analogues. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
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Abstract In this article we study the homology of spaces $$\mathrm{Hom}({\mathbb Z}^n,G)$$ 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 $$\mathrm{Hom}(F_n/\Gamma _n^m,G)$$, where the subgroups $$\Gamma _n^m$$ 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 $$\mathrm{Comm}(G)$$ studied by Cohen–Stafa and its nilpotent analogues. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
| abstract_unstemmed |
Abstract In this article we study the homology of spaces $$\mathrm{Hom}({\mathbb Z}^n,G)$$ 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 $$\mathrm{Hom}(F_n/\Gamma _n^m,G)$$, where the subgroups $$\Gamma _n^m$$ 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 $$\mathrm{Comm}(G)$$ studied by Cohen–Stafa and its nilpotent analogues. © Springer-Verlag GmbH Germany, part of Springer Nature 2018 |
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Hilbert–Poincaré series for spaces of commuting elements in Lie groups |
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