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

 Algebra Discrete Math., 2016, Volume 22, Issue 2, Pages 240–261 (Mi adm586)

RESEARCH ARTICLE

A horizontal mesh algorithm for posets with positive Tits form

Mariusz Kaniecki, Justyna Kosakowska, Piotr Malicki, Grzegorz Marczak

Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland

Abstract: Following our paper [Fund. Inform. 136 (2015), 345–379], we define a horizontal mesh algorithm that constructs a $\widehat{\Phi}_I$-mesh translation quiver $\Gamma(\widehat{\mathcal{R}}_I,\widehat{\Phi}_I)$ consisting of $\widehat{\Phi}_I$-orbits of the finite set $\widehat{\mathcal{R}}_I=\{v\in\mathbb{Z}^I\; ;\;\widehat{q}_I(v)=1\}$ of Tits roots of a poset $I$ with positive definite Tits quadratic form $\widehat q_I:\mathbb{Z}^I \to \mathbb{Z}$. Under the assumption that $\widehat q_I:\mathbb{Z}^I \to \mathbb{Z}$ is positive definite, the algorithm constructs $\Gamma(\widehat{\mathcal{R}}_I,\widehat{\Phi}_I)$ such that it is isomorphic with the $\widehat{\Phi}_D$-mesh translation quiver $\Gamma({\mathcal{R}}_D,{\Phi}_D)$ of $\widehat{\Phi}_D$-orbits of the finite set ${\mathcal{R}}_D$ of roots of a simply laced Dynkin quiver $D$ associated with $I$.

Keywords: poset, combinatorial algorithm, Dynkin diagram, mesh geometry of roots, quadratic form.

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Bibliographic databases:
MSC: 68R10, 05C50, 06A07, 15A63
Revised: 05.01.2016
Language:

Citation: Mariusz Kaniecki, Justyna Kosakowska, Piotr Malicki, Grzegorz Marczak, “A horizontal mesh algorithm for posets with positive Tits form”, Algebra Discrete Math., 22:2 (2016), 240–261

Citation in format AMSBIB
\Bibitem{KanKosMal16} \by Mariusz~Kaniecki, Justyna~Kosakowska, Piotr~Malicki, Grzegorz~Marczak \paper A~horizontal mesh algorithm for posets with~positive Tits form \jour Algebra Discrete Math. \yr 2016 \vol 22 \issue 2 \pages 240--261 \mathnet{http://mi.mathnet.ru/adm586} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3593123} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000392709600007}