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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2014, Volume 17, Issue 1, Pages 33–69 (Mi adm458)

RESEARCH ARTICLE

Algorithmic computation of principal posets using Maple and Python

Marcin Gąsiorek, Daniel Simson, Katarzyna Zając

Faculty of Mathematics and Computer, Science, Nicolaus Copernicus University, 87-100 Toruń, Poland

Abstract: 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 $G_I : = \frac{1}{2}[C_I+ C^{tr}_I]\in\mathbb{M}_I(\mathbb{Q})$ of $I$ is positive semi-definite of corank one, where $C_I\in\mathbb{M}_I(\mathbb{Z})$ is the incidence matrix of $I$. With any such a connected poset $I$, we associate a simply laced Euclidean diagram $DI\in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n, \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}$, the Coxeter matrix $Cox_I:= - C_I\cdot C^{-tr}_I$, its complex Coxeter spectrum ${\mathbf{specc}}_I$, and a reduced Coxeter number $\check {\mathbf{c}}_I$. One of our aims is to show that the spectrum ${\mathbf{specc}}_I$ of any such a poset $I$ determines the incidence matrix $C_I$ (hence the poset $I$) uniquely, up to a $\mathbb{Z}$-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 $\leq 15$, together with ${\mathbf{specc}}_I$, $\check {\mathbf{c}}_I$, the incidence defect $\partial_I:\mathbb{Z}^I \to\mathbb{Z}$, and the Coxeter-Euclidean type $DI$. In case when $DI \in \{\widetilde{\mathbb{A}}_n, \widetilde{\mathbb{D}}_n , \widetilde{\mathbb{E}}_6, \widetilde{\mathbb{E}}_7, \widetilde{\mathbb{E}}_8\}$ and $n:=|I|$ is relatively small, we show that given such a principal poset $I$, the incidence matrix $C_I$ is $\mathbb{Z}$-congruent with the non-symmetric Gram matrix $\check G_{DI}$ of $DI$, ${\mathbf{specc}}_I = {\mathbf{specc}}_{DI}$ and $\check {\mathbf{c}}_I= \check {\mathbf{c}}_{DI}$. Moreover, given a pair of principal posets $I$ and $J$, with $|I|= |J| \leq 15$, the matrices $C_I$ and $C_J$ are $\mathbb{Z}$-congruent if and only if ${\mathbf{specc}}_I= {\mathbf{specc}}_J$.

Keywords: principal poset; edge-bipartite graph; unit quadratic form; computer algorithm; Gram matrix, Coxeter polynomial, Coxeter spectrum.

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Bibliographic databases:
MSC: 06A11, 15A63, 68R05, 68W30
Revised: 08.08.2013
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Citation: Marcin Gąsiorek, Daniel Simson, Katarzyna Zając, “Algorithmic computation of principal posets using Maple and Python”, Algebra Discrete Math., 17:1 (2014), 33–69

Citation in format AMSBIB
\Bibitem{GasSimZaj14} \by Marcin~G{\k a}siorek, Daniel~Simson, Katarzyna~Zaj{\k a}c \paper Algorithmic computation of principal posets using Maple and Python \jour Algebra Discrete Math. \yr 2014 \vol 17 \issue 1 \pages 33--69 \mathnet{http://mi.mathnet.ru/adm458} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3288184} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000352198400003} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Polak A., Simson D., “Coxeter Spectral Classification of Almost Tp-Critical One-Peak Posets Using Symbolic and Numeric Computations”, Linear Alg. Appl., 445 (2014), 223–255
2. Marczak G., Simson D., Zajac K., “on Computing Non-Negative Loop-Free Edge-Bipartite Graphs”, 2013 15Th International Symposium on Symbolic and Numeric Algorithms For Scientific Computing (Synasc 2013), eds. Bjorner N., Negru V., Ida T., Jebelean T., Petcu D., Watt S., Zaharie D., IEEE, 2014, 68–75
3. Gasiorek M. Simson D. Zajac K., “On Corank Two Edge-Bipartite Graphs and Simply Extended Euclidean Diagrams”, 16Th International Symposium on Symbolic and Numeric Algorithms For Scientific Computing (Synasc 2014), ed. Winkler F. Negru V. Ida T. Jebelean T. Petcu D. Watt S. Zaharie D., IEEE Computer Soc, 2014, 66–73
4. Gasiorek M., “Efficient Computation of the Isotropy Group of a Finite Graph: a Combinatorial Approach”, 2013 15Th International Symposium on Symbolic and Numeric Algorithms For Scientific Computing (Synasc 2013), International Symposium on Symbolic and Numeric Algorithms For Scientific Computing, eds. Bjorner N., Negru V., Ida T., Jebelean T., Petcu D., Watt S., Zaharie D., IEEE, 2014, 104–111
5. Felisiak M., Simson D., “Applications of Matrix Morsifications To Coxeter Spectral Study of Loop-Free Edge-Bipartite Graphs”, Discrete Appl. Math., 192:SI (2015), 49–64
6. Gasiorek M., Simson D., Zajac K., “on Coxeter Type Study of Non-Negative Posets Using Matrix Morsifications and Isotropy Groups of Dynkin and Euclidean Diagrams”, Eur. J. Comb., 48:SI (2015), 127–142
7. Gasiorek M. Simson D. Zajac K., “Structure and a Coxeter-Dynkin Type Classification of Corank Two Non-Negative Posets”, Linear Alg. Appl., 469 (2015), 76–113
8. Kasjan S., Simson D., “Mesh Algorithms For Coxeter Spectral Classification of Cox-Regular Edge-Bipartite Graphs With Loops, i. Mesh Root Systems”, Fundam. Inform., 139:2 (2015), 153–184
9. Kasjan S., Simson D., “Mesh Algorithms For Coxeter Spectral Classification of Cox-Regular Edge-Bipartite Graphs With Loops, II. Application To Coxeter Spectral Analysis”, Fundam. Inform., 139:2 (2015), 185–209
10. Kasjan S., Simson D., “Algorithms For Isotropy Groups of Cox-Regular Edge-Bipartite Graphs”, Fundam. Inform., 139:3 (2015), 249–275
11. M. Gasiorek, D. Simson, K. Zajac, “A Gram classification of non-negative corank-two loop-free edge-bipartite graphs”, Linear Alg. Appl., 500 (2016), 88–118
12. D. Simson, “Symbolic algorithms computing Gram congruences in the Coxeter spectral classification of edge-bipartite graphs, I. A Gram classification”, Fundam. Inform., 145:1 (2016), 19–48
13. D. Simson, “Symbolic algorithms computing Gram congruences in the Coxeter spectral classification of edge-bipartite graphs, II. Isotropy mini-groups”, Fundam. Inform., 145:1 (2016), 49–80
14. D. Simson, K. Zajac, “Inflation algorithm for loop-free non-negative edge-bipartite graphs of corank at least two”, Linear Alg. Appl., 524 (2017), 109–152
15. K. Zajac, “Numeric algorithms for corank two edge-bipartite graphs and their mesh geometries of roots”, Fundam. Inform., 152:2 (2017), 185–222
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