The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains
Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are...
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| Autor*in: |
Chapovalov, Maxim [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2010 |
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| Schlagwörter: |
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| Anmerkung: |
© M. Chapovalov, D. Leites and R. Stekolshchik 2010 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of nonlinear mathematical physics - Abingdon, Oxon : Taylor & Francis, 1994, 17(2010), Suppl 1 vom: Jan., Seite 169-215 |
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| Übergeordnetes Werk: |
volume:17 ; year:2010 ; number:Suppl 1 ; month:01 ; pages:169-215 |
| Links: |
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| DOI / URN: |
10.1142/S1402925110000842 |
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SPR054468779 |
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| 520 | |a Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. | ||
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| 700 | 1 | |a Stekolshchik, Rafael |4 aut | |
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10.1142/S1402925110000842 doi (DE-627)SPR054468779 (SPR)S1402925110000842-e DE-627 ger DE-627 rakwb eng Chapovalov, Maxim verfasserin aut The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © M. Chapovalov, D. Leites and R. Stekolshchik 2010 Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. Hilbert-Poincaré series (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 Leites, Dimitry aut Stekolshchik, Rafael aut Enthalten in Journal of nonlinear mathematical physics Abingdon, Oxon : Taylor & Francis, 1994 17(2010), Suppl 1 vom: Jan., Seite 169-215 (DE-627)325293635 (DE-600)2034956-7 1776-0852 nnns volume:17 year:2010 number:Suppl 1 month:01 pages:169-215 https://dx.doi.org/10.1142/S1402925110000842 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 17 2010 Suppl 1 01 169-215 |
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10.1142/S1402925110000842 doi (DE-627)SPR054468779 (SPR)S1402925110000842-e DE-627 ger DE-627 rakwb eng Chapovalov, Maxim verfasserin aut The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © M. Chapovalov, D. Leites and R. Stekolshchik 2010 Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. Hilbert-Poincaré series (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 Leites, Dimitry aut Stekolshchik, Rafael aut Enthalten in Journal of nonlinear mathematical physics Abingdon, Oxon : Taylor & Francis, 1994 17(2010), Suppl 1 vom: Jan., Seite 169-215 (DE-627)325293635 (DE-600)2034956-7 1776-0852 nnns volume:17 year:2010 number:Suppl 1 month:01 pages:169-215 https://dx.doi.org/10.1142/S1402925110000842 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 17 2010 Suppl 1 01 169-215 |
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10.1142/S1402925110000842 doi (DE-627)SPR054468779 (SPR)S1402925110000842-e DE-627 ger DE-627 rakwb eng Chapovalov, Maxim verfasserin aut The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © M. Chapovalov, D. Leites and R. Stekolshchik 2010 Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. Hilbert-Poincaré series (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 Leites, Dimitry aut Stekolshchik, Rafael aut Enthalten in Journal of nonlinear mathematical physics Abingdon, Oxon : Taylor & Francis, 1994 17(2010), Suppl 1 vom: Jan., Seite 169-215 (DE-627)325293635 (DE-600)2034956-7 1776-0852 nnns volume:17 year:2010 number:Suppl 1 month:01 pages:169-215 https://dx.doi.org/10.1142/S1402925110000842 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 17 2010 Suppl 1 01 169-215 |
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10.1142/S1402925110000842 doi (DE-627)SPR054468779 (SPR)S1402925110000842-e DE-627 ger DE-627 rakwb eng Chapovalov, Maxim verfasserin aut The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © M. Chapovalov, D. Leites and R. Stekolshchik 2010 Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. Hilbert-Poincaré series (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 Leites, Dimitry aut Stekolshchik, Rafael aut Enthalten in Journal of nonlinear mathematical physics Abingdon, Oxon : Taylor & Francis, 1994 17(2010), Suppl 1 vom: Jan., Seite 169-215 (DE-627)325293635 (DE-600)2034956-7 1776-0852 nnns volume:17 year:2010 number:Suppl 1 month:01 pages:169-215 https://dx.doi.org/10.1142/S1402925110000842 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 17 2010 Suppl 1 01 169-215 |
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10.1142/S1402925110000842 doi (DE-627)SPR054468779 (SPR)S1402925110000842-e DE-627 ger DE-627 rakwb eng Chapovalov, Maxim verfasserin aut The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains 2010 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © M. Chapovalov, D. Leites and R. Stekolshchik 2010 Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. Hilbert-Poincaré series (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 Leites, Dimitry aut Stekolshchik, Rafael aut Enthalten in Journal of nonlinear mathematical physics Abingdon, Oxon : Taylor & Francis, 1994 17(2010), Suppl 1 vom: Jan., Seite 169-215 (DE-627)325293635 (DE-600)2034956-7 1776-0852 nnns volume:17 year:2010 number:Suppl 1 month:01 pages:169-215 https://dx.doi.org/10.1142/S1402925110000842 kostenfrei Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_2014 GBV_ILN_2088 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 17 2010 Suppl 1 01 169-215 |
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The Poincaré Series of the Hyperbolic Coxeter Groups with Finite Volume of Fundamental Domains |
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Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. © M. Chapovalov, D. Leites and R. Stekolshchik 2010 |
| abstractGer |
Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. © M. Chapovalov, D. Leites and R. Stekolshchik 2010 |
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Abstract The discrete group generated by reflections of the sphere, or the Euclidean space, or hyperbolic space are said to be Coxeter groups of, respectively, spherical, or Euclidean, or hyperbolic type. The hyperbolic Coxeter groups are said to be (quasi-)Lannér if the tiles covering the space are of finite volume and all (resp. some of them) are compact. For any Coxeter group stratified by the length of its elements, the Poincaré series is the generating function of the cardinalities of sets of elements of equal length. Around 1966, Solomon established that, for ANY Coxeter group, its Poincaré series is a rational function with zeros somewhere on the unit circle centered at the origin, and gave an implicit (recurrence) formula. For the spherical and Euclidean Coxeter groups, the explicit expression of the Poincaré series is well-known. The explicit answer was known for any 3-generated Coxeter group, and (with mistakes) for the Lannér groups. Here we give a lucid description of the numerator of the Poincaré series of any Coxeter group, the explicit expression of the Poincaré series for each Lannér and quasi-Lannér group, and review the scene. We give an interpretation of some coefficients of the denominator of the growth function. The non-real poles behave as in Eneström’s theorem (lie in a narrow annulus) though the coefficients of the denominators do not satisfy theorem’s requirements. © M. Chapovalov, D. Leites and R. Stekolshchik 2010 |
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