Subalgebras of the Fomin–Kirillov algebra
Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Blasiak, Jonah [verfasserIn] Liu, Ricky Ini [verfasserIn] Mészáros, Karola [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
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| Übergeordnetes Werk: |
Enthalten in: Journal of algebraic combinatorics - Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992, 44(2016), 3 vom: 11. Mai, Seite 785-829 |
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| Übergeordnetes Werk: |
volume:44 ; year:2016 ; number:3 ; day:11 ; month:05 ; pages:785-829 |
| Links: |
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| DOI / URN: |
10.1007/s10801-016-0688-4 |
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| Katalog-ID: |
SPR013326384 |
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| 245 | 1 | 0 | |a Subalgebras of the Fomin–Kirillov algebra |
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| 520 | |a Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. | ||
| 650 | 4 | |a Fomin–Kirillov algebra |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Coxeter group |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a nil-Coxeter algebra |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Nichols algebra |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Liu, Ricky Ini |e verfasserin |4 aut | |
| 700 | 1 | |a Mészáros, Karola |e verfasserin |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Journal of algebraic combinatorics |d Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992 |g 44(2016), 3 vom: 11. Mai, Seite 785-829 |w (DE-627)271180722 |w (DE-600)1479499-8 |x 1572-9192 |7 nnns |
| 773 | 1 | 8 | |g volume:44 |g year:2016 |g number:3 |g day:11 |g month:05 |g pages:785-829 |
| 856 | 4 | 0 | |u https://dx.doi.org/10.1007/s10801-016-0688-4 |z lizenzpflichtig |3 Volltext |
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10.1007/s10801-016-0688-4 doi (DE-627)SPR013326384 (SPR)s10801-016-0688-4-e DE-627 ger DE-627 rakwb eng 510 ASE 31.12 bkl 31.20 bkl Blasiak, Jonah verfasserin aut Subalgebras of the Fomin–Kirillov algebra 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. Fomin–Kirillov algebra (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 nil-Coxeter algebra (dpeaa)DE-He213 Nichols algebra (dpeaa)DE-He213 Liu, Ricky Ini verfasserin aut Mészáros, Karola verfasserin aut Enthalten in Journal of algebraic combinatorics Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992 44(2016), 3 vom: 11. Mai, Seite 785-829 (DE-627)271180722 (DE-600)1479499-8 1572-9192 nnns volume:44 year:2016 number:3 day:11 month:05 pages:785-829 https://dx.doi.org/10.1007/s10801-016-0688-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.12 ASE 31.20 ASE AR 44 2016 3 11 05 785-829 |
| spelling |
10.1007/s10801-016-0688-4 doi (DE-627)SPR013326384 (SPR)s10801-016-0688-4-e DE-627 ger DE-627 rakwb eng 510 ASE 31.12 bkl 31.20 bkl Blasiak, Jonah verfasserin aut Subalgebras of the Fomin–Kirillov algebra 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. Fomin–Kirillov algebra (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 nil-Coxeter algebra (dpeaa)DE-He213 Nichols algebra (dpeaa)DE-He213 Liu, Ricky Ini verfasserin aut Mészáros, Karola verfasserin aut Enthalten in Journal of algebraic combinatorics Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992 44(2016), 3 vom: 11. Mai, Seite 785-829 (DE-627)271180722 (DE-600)1479499-8 1572-9192 nnns volume:44 year:2016 number:3 day:11 month:05 pages:785-829 https://dx.doi.org/10.1007/s10801-016-0688-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.12 ASE 31.20 ASE AR 44 2016 3 11 05 785-829 |
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10.1007/s10801-016-0688-4 doi (DE-627)SPR013326384 (SPR)s10801-016-0688-4-e DE-627 ger DE-627 rakwb eng 510 ASE 31.12 bkl 31.20 bkl Blasiak, Jonah verfasserin aut Subalgebras of the Fomin–Kirillov algebra 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. Fomin–Kirillov algebra (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 nil-Coxeter algebra (dpeaa)DE-He213 Nichols algebra (dpeaa)DE-He213 Liu, Ricky Ini verfasserin aut Mészáros, Karola verfasserin aut Enthalten in Journal of algebraic combinatorics Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992 44(2016), 3 vom: 11. Mai, Seite 785-829 (DE-627)271180722 (DE-600)1479499-8 1572-9192 nnns volume:44 year:2016 number:3 day:11 month:05 pages:785-829 https://dx.doi.org/10.1007/s10801-016-0688-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.12 ASE 31.20 ASE AR 44 2016 3 11 05 785-829 |
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10.1007/s10801-016-0688-4 doi (DE-627)SPR013326384 (SPR)s10801-016-0688-4-e DE-627 ger DE-627 rakwb eng 510 ASE 31.12 bkl 31.20 bkl Blasiak, Jonah verfasserin aut Subalgebras of the Fomin–Kirillov algebra 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. Fomin–Kirillov algebra (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 nil-Coxeter algebra (dpeaa)DE-He213 Nichols algebra (dpeaa)DE-He213 Liu, Ricky Ini verfasserin aut Mészáros, Karola verfasserin aut Enthalten in Journal of algebraic combinatorics Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992 44(2016), 3 vom: 11. Mai, Seite 785-829 (DE-627)271180722 (DE-600)1479499-8 1572-9192 nnns volume:44 year:2016 number:3 day:11 month:05 pages:785-829 https://dx.doi.org/10.1007/s10801-016-0688-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.12 ASE 31.20 ASE AR 44 2016 3 11 05 785-829 |
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10.1007/s10801-016-0688-4 doi (DE-627)SPR013326384 (SPR)s10801-016-0688-4-e DE-627 ger DE-627 rakwb eng 510 ASE 31.12 bkl 31.20 bkl Blasiak, Jonah verfasserin aut Subalgebras of the Fomin–Kirillov algebra 2016 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. Fomin–Kirillov algebra (dpeaa)DE-He213 Coxeter group (dpeaa)DE-He213 nil-Coxeter algebra (dpeaa)DE-He213 Nichols algebra (dpeaa)DE-He213 Liu, Ricky Ini verfasserin aut Mészáros, Karola verfasserin aut Enthalten in Journal of algebraic combinatorics Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992 44(2016), 3 vom: 11. Mai, Seite 785-829 (DE-627)271180722 (DE-600)1479499-8 1572-9192 nnns volume:44 year:2016 number:3 day:11 month:05 pages:785-829 https://dx.doi.org/10.1007/s10801-016-0688-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2116 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4012 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 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_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.12 ASE 31.20 ASE AR 44 2016 3 11 05 785-829 |
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Enthalten in Journal of algebraic combinatorics 44(2016), 3 vom: 11. Mai, Seite 785-829 volume:44 year:2016 number:3 day:11 month:05 pages:785-829 |
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Fomin–Kirillov algebra Coxeter group nil-Coxeter algebra Nichols algebra |
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Blasiak, Jonah @@aut@@ Liu, Ricky Ini @@aut@@ Mészáros, Karola @@aut@@ |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR013326384</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220111002053.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201006s2016 xx |||||o 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10801-016-0688-4</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR013326384</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s10801-016-0688-4-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">31.12</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">31.20</subfield><subfield code="2">bkl</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Blasiak, Jonah</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Subalgebras of the Fomin–Kirillov algebra</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2016</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Fomin–Kirillov algebra</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Coxeter group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">nil-Coxeter algebra</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nichols algebra</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Liu, Ricky Ini</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Mészáros, Karola</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Journal of algebraic combinatorics</subfield><subfield code="d">Dordrecht [u.a.] : Springer Science + Business Media B.V., 1992</subfield><subfield code="g">44(2016), 3 vom: 11. 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Blasiak, Jonah |
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subalgebras of the fomin–kirillov algebra |
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Subalgebras of the Fomin–Kirillov algebra |
| abstract |
Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. |
| abstractGer |
Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. |
| abstract_unstemmed |
Abstract The Fomin–Kirillov algebra %${\mathcal {E}}_n%$ is a noncommutative quadratic algebra with a generator for every edge of the complete graph on n vertices. For any graph G on n vertices, we define %${{\mathcal {E}}_G}%$ to be the subalgebra of %${\mathcal {E}}_n%$ generated by the edges of G. We show that these algebras have many parallels with Coxeter groups and their nil-Coxeter algebras: for instance, %${\mathcal {E}}_G%$ is a free %${\mathcal {E}}_H%$-module for any %$H\subseteq G%$, and if %${\mathcal {E}}_G%$ is finite-dimensional, then its Hilbert series has symmetric coefficients. We determine explicit monomial bases and Hilbert series for %${\mathcal {E}}_G%$ when G is a simply laced finite Dynkin diagram or a cycle, in particular showing that %${\mathcal {E}}_G%$ is finite-dimensional in these cases. We also present conjectures for the Hilbert series of %${\mathcal {E}}_{\tilde{D}_n}%$, %${\mathcal {E}}_{\tilde{E}_6}%$, and %${\mathcal {E}}_{\tilde{E}_7}%$, as well as the graphs G on six vertices for which %$\mathcal {E}_G%$ is finite-dimensional. |
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| container_issue |
3 |
| title_short |
Subalgebras of the Fomin–Kirillov algebra |
| url |
https://dx.doi.org/10.1007/s10801-016-0688-4 |
| remote_bool |
true |
| author2 |
Liu, Ricky Ini Mészáros, Karola |
| author2Str |
Liu, Ricky Ini Mészáros, Karola |
| ppnlink |
271180722 |
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c |
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false |
| doi_str |
10.1007/s10801-016-0688-4 |
| up_date |
2024-07-03T18:56:31.042Z |
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1803585313621147648 |
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| score |
7.401602 |