Algorithmic computation of principal posets using Maple and Python
Vernadsky National Library of Ukraine
Переглянути архів Інформація| Поле | Співвідношення | |
| Title |
Algorithmic computation of principal posets using Maple and Python
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| Creator |
Gasiorek, M.
Simson, D. Zajac, K. |
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| 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.
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| Date |
2019-06-10T10:53:58Z
2019-06-10T10:53:58Z 2014 |
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| Type |
Article
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| Identifier |
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 |
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| Language |
en
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| Relation |
Algebra and Discrete Mathematics
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| Publisher |
Інститут прикладної математики і механіки НАН України
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