Библиографическое описание:О СВЯЗИ НЕКОТОРЫХ ГРУПП, ПОРОЖДЕННЫХ 3-ТРАНСПОЗИЦИЯМИ, С ГРУППАМИ КОКСТЕРА = On the connection of some groups generated by 3-transpositions with Coxeter groups : научное издание / В. М. Синицин, А. И. Созутов. - Текст : непосредственный // Труды института математики и механики УрО РАН = TRUDY INSTITUTA MATEMATIKI I MEKHANIKI URO RAN. - 2020. - Т. 26, № 4. - С. 234-243. - Cited Reference Count: 17. - This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00566 A.), Russian Foundation for Basic ResearchRussian Foundation for Basic Research (RFBR) [19-01-00566 A]. - ISSN 0134-4889. - ISSN 2658-4786, DOI 10.21538/0134-4889-2020-26-4-234-243.
Аннотация:Группы Кокстера, более известные как группы, порожденные отражениями, имеют многочисленные приложения в различных областях математики и за ее пределами. Группы с 3-транспозициями Фишера также связаны со многими структурами: конечные простые группы, тройные графы, геометрии различных пространств, алгебры Ли и др. Пересечение этих классов групп состоит из конечных групп Вейля $W(A_n)\simeq S_{n+1}$, $W(D_n)$, $W(E_n)$ ($n=6,7,8$) простых конечномерных алгебр и групп Ли. В работе продолжается исследование связи между конечными группами $Sp_{2l}(2)$ и $O
Coxeter groups, more commonly known as reflection-generated groups, have numerous applications in various fields of mathematics and beyond. Groups with Fischer's 3-transpositions are also related to many structures: finite simple groups, triple graphs, geometries of various spaces, Lie algebras, etc. The intersection of these classes of groups consists of finite Weyl groups $W(A_n)\simeq S_{n+1}$, $W(D_n)$, and $W(E_n)$ ($n=6,7,8$) of simple finite-dimensional algebras and Lie groups. The paper continues the study of the connection between the finite groups $Sp_{2l}(2)$ and $O
Coxeter groups, more commonly known as reflection-generated groups, have numerous applications in various fields of mathematics and beyond. Groups with Fischer's 3-transpositions are also related to many structures: finite simple groups, triple graphs, geometries of various spaces, Lie algebras, etc. The intersection of these classes of groups consists of finite Weyl groups W(A(n)) similar or equal to Sn+1, W(D-n), and W(E-n) (n = 6, 7, 8) of simple finite-dimensional algebras and Lie groups. The paper continues the study of the connection between the finite groups Sp(2l)(2) and O-2l(+/-) (2) from clauses (ii)-(iii) of Fischer's theorem and infinite Coxeter groups. The organizing basis of the connection under study is general Coxeter tree graphs Gamma(n) with vertices 1, ... , n. To each vertex i of the graph Gamma(n), we assign the generating involution (reflection) s(i) of the Coxeter group G(n), the basis vector e(i) of the space V-n over the field F-2 of two elements, and the generating transvection w(i) of the subgroup W-n = of SL(V-n) = SLn(2). The graph Gamma(n) corresponds to exactly one Coxeter group of rank n: G(n) = , where m(ii) = 1, 1 w(i) (i = 1, ... , n) is continued to the surjective homomorphism G(n) -> W-n. In the authors' previous paper, for all groups W-n = O-2l(+/-)(2) (n = 2l >= 6) and W-n = Sp(2l)(2) (n = 2l + 1 >= 7), an algorithm was given for enumerating the corresponding tree graphs Gamma(n) by grouping them according to E-series of nested graphs. In the present paper, a close genetic connection is established between the groups O-2l(+/-)(2) and Sp(2l)(2) x Z(2) (3
Ключевые слова:groups with 3-transpositions, Coxeter graphs and groups, genetic codes, группы с 3-транспозициями, графы и группы Кокстера, генетические коды
