Algorithmic computation of principal posets using Maple and Python
We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets I that are principal, i.e., the rational symmetric Gram matrix GI : = 1/2[CI+CItr] ∈ MI(Q) of I is positive semi-definite of...
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| Дата: | 2014 |
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| Автори: | , , |
| Формат: | Стаття |
| Мова: | English |
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Інститут прикладної математики і механіки НАН України
2014
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| Назва видання: | Algebra and Discrete Mathematics |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/152339 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Algorithmic computation of principal posets using Maple and Python / M. Gasiorek, D. Simson, K. Zajac // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 1. — С. 33–69. — Бібліогр.: 56 назв. — англ. |
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irk-123456789-1523392019-06-11T01:25:16Z Algorithmic computation of principal posets using Maple and Python Gasiorek, M. Simson, D. Zajac, K. We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets I that are principal, i.e., the rational symmetric Gram matrix GI : = 1/2[CI+CItr] ∈ MI(Q) of I is positive semi-definite of corank one, where CI ∈ MI(Z) is the incidence matrix of I. With any such a connected poset I, we associate a simply laced Euclidean diagram DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈}, the Coxeter matrix CoxI := −CI ⋅ C−trI, its complex Coxeter spectrum speccI, and a reduced Coxeter number cI. One of our aims is to show that the spectrum speccI of any such a poset I determines the incidence matrix CI (hence the poset I) uniquely, up to a Z-congruence. By computer calculations, we find a complete list of principal one-peak posets I (i.e., I has a unique maximal element) of cardinality ≤ 15, together with speccI, cI, the incidence defect ∂I : ZI → Z, and the Coxeter-Euclidean type DI. In case when DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈} and n := |I| is relatively small, we show that given such a principal poset I, the incidence matrix CI is Z-congruent with the non-symmetric Gram matrix GˇDI of DI, speccI = speccDI and cˇI = cˇDI. Moreover, given a pair of principal posets I and J, with |I| = |J| ≤ 15, the matrices CI and CJ are Z-congruent if and only if speccI = speccJ. 2014 Article Algorithmic computation of principal posets using Maple and Python / M. Gasiorek, D. Simson, K. Zajac // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 1. — С. 33–69. — Бібліогр.: 56 назв. — англ. 1726-3255 2010 MSC:06A11, 15A63, 68R05, 68W30. http://dspace.nbuv.gov.ua/handle/123456789/152339 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
We present symbolic and numerical algorithms for a computer search in the Coxeter spectral classification problems. One of the main aims of the paper is to study finite posets I that are principal, i.e., the rational symmetric Gram matrix GI : = 1/2[CI+CItr] ∈ MI(Q) of I is positive semi-definite of corank one, where CI ∈ MI(Z) is the incidence matrix of I. With any such a connected poset I, we associate a simply laced Euclidean diagram DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈}, the Coxeter matrix CoxI := −CI ⋅ C−trI, its complex Coxeter spectrum speccI, and a reduced Coxeter number cI. One of our aims is to show that the spectrum speccI of any such a poset I determines the incidence matrix CI (hence the poset I) uniquely, up to a Z-congruence. By computer calculations, we find a complete list of principal one-peak posets I (i.e., I has a unique maximal element) of cardinality ≤ 15, together with speccI, cI, the incidence defect ∂I : ZI → Z, and the Coxeter-Euclidean type DI. In case when DI ∈ {A˜n, D˜n, E˜₆, E˜₇, E˜₈} and n := |I| is relatively small, we show that given such a principal poset I, the incidence matrix CI is Z-congruent with the non-symmetric Gram matrix GˇDI of DI, speccI = speccDI and cˇI = cˇDI. Moreover, given a pair of principal posets I and J, with |I| = |J| ≤ 15, the matrices CI and CJ are Z-congruent if and only if speccI = speccJ. |
| format |
Article |
| author |
Gasiorek, M. Simson, D. Zajac, K. |
| spellingShingle |
Gasiorek, M. Simson, D. Zajac, K. Algorithmic computation of principal posets using Maple and Python Algebra and Discrete Mathematics |
| author_facet |
Gasiorek, M. Simson, D. Zajac, K. |
| author_sort |
Gasiorek, M. |
| title |
Algorithmic computation of principal posets using Maple and Python |
| title_short |
Algorithmic computation of principal posets using Maple and Python |
| title_full |
Algorithmic computation of principal posets using Maple and Python |
| title_fullStr |
Algorithmic computation of principal posets using Maple and Python |
| title_full_unstemmed |
Algorithmic computation of principal posets using Maple and Python |
| title_sort |
algorithmic computation of principal posets using maple and python |
| publisher |
Інститут прикладної математики і механіки НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/152339 |
| citation_txt |
Algorithmic computation of principal posets using Maple and Python / M. Gasiorek, D. Simson, K. Zajac // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 1. — С. 33–69. — Бібліогр.: 56 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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