RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Chebyshevskii Sb.: Year: Volume: Issue: Page: Find

 Chebyshevskii Sb., 2016, Volume 17, Issue 2, Pages 113–127 (Mi cheb482)

On normalizers in some Coxeter groups

I. V. Dobrynina

Tula State Pedagogical University

Abstract: Let $G$ be a finitely generated Coxeter group with presentation
$$G=< a_1,\ldots, a_n;(a_ia_j)^{m_{ij}}=1, i,j =\overline{1,n} >,$$
where $m_{ij}$ — are the elements of the symmetric Coxeter matrix: $\forall i,j \in\overline{1,n}, m_{ii}=1, m_{ij} \geq$ $\geq2, i\ne j$.
If $m_{ij}\geq3$ $(m_{ij}>3)$, $i\ne j$, then $G$ is a Coxeter group of large (extra-large) type. These groups introduced by K. Appel and P. Schupp.
If the group $G$ corresponds to a finite tree-graph $\Gamma$ such that if the vertices of some edge $e$ of the graph $\Gamma$ correspond to generators $a_i, a_j$, then the edge $e$ corresponds to the ratio of the species $(a_ia_j)^{m_{ij}}=1$, then $G$ is a Coxeter group with a tree-structure.
Coxeter groups with a tree-structure introduced by V. N. Bezverkhnii, algorithmic problems in them was considered by V. N. Bezverkhnii and O. V. Inchenko.
The group $G$ can be represented as tree product 2-generated of Coxeter groups, amalgamated by cyclic subgroups. Thus from the graph $\Gamma$ of $G$ will move to the graph $\overline{\Gamma}$ in the following way: the vertices of the graph $\overline{\Gamma}$ we will put in line Coxeter group on two generators
$$G_{ij} = <a_i, a_j; a_i^2=a_j^2=1,(a_ia_j)^{m_{ij}}=1>$$
and
$$G_{jk} = <a_j, a_k; a_j^2=a_k^2=1,(a_ja_k)^{m_{jk}}=1>,$$
to every edge $\overline{e}$ joining the vertices corresponding to $G_{ij}$ and $G_{jk}$ is a cyclic subgroup
$$<a_j;a_j^2=1>.$$

In this paper we prove the following theorem: normalizer of finitely generated subgroup of Coxeter group with tree-structure
$$\overline{G}=G_{ij}\ast_{<a_j; a_j^2>}G_{jk},$$

$$G_{ij} = <a_i, a_j; a_i^2=a_j^2=1,(a_ia_j)^{m_{ij}}=1>,$$

$$G_{jk} = <a_j, a_k; a_j^2=a_k^2=1,(a_ja_k)^{m_{jk}}=1>$$
finitely generated and exist algorithm for generating.
Bibliography: 18 titles.

Keywords: Coxeter group, tree-structure, normalizer, amalgamated product.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-41-03222_ð_öåíòð_à

Full text: PDF file (662 kB)
References: PDF file   HTML file
UDC: 519.4
Accepted:10.06.2016

Citation: I. V. Dobrynina, “On normalizers in some Coxeter groups”, Chebyshevskii Sb., 17:2 (2016), 113–127

Citation in format AMSBIB
\Bibitem{Dob16} \by I.~V.~Dobrynina \paper On normalizers in some Coxeter groups \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 2 \pages 113--127 \mathnet{http://mi.mathnet.ru/cheb482} \elib{https://elibrary.ru/item.asp?id=26254427} 

• http://mi.mathnet.ru/eng/cheb482
• http://mi.mathnet.ru/eng/cheb/v17/i2/p113

 SHARE:

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. V. N. Bezverkhnii, N. B. Bezverkhnyaya, I. V. Dobrynina, O. V. Inchenko, A. E. Ustyan, “Ob algoritmicheskikh problemakh v gruppakh Kokstera”, Chebyshevskii sb., 17:4 (2016), 23–50
2. I. V. Dobrynina, “O normalizatorakh podgrupp v gruppakh Kokstera s drevesnoi strukturoi”, Sib. elektron. matem. izv., 14 (2017), 1338–1348
•  Number of views: This page: 106 Full text: 38 References: 13