On the connection of some groups generated by 3-transpositions with Coxeter groups
V. M. Sinitsin,
A. I. Sozutov Siberian Federal University, Krasnoyarsk
Abstract:
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
$O^\pm_{2l}(2)$ 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
$1,\ldots, n$. 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
$W_n=\langle w_1,\ldots,w_n\rangle$ of
$SL(V_n)=SL_n(2)$. The graph
$\Gamma_n$ corresponds to exactly one Coxeter group of rank
$n$: $G_n=\langle s_1,\ldots,s_n\mid (s_is_j)^{m_{ij}},\, m_{ij}\leq 3\rangle$, where
$m_{ii}=1$,
$1\leq i<j\leq n$, and
$m_{ij}=3$ or
$m_{ij}=2$ depending on whether
$\Gamma_n$ contains the edge
$(i,j)$. The form defined by the graph
$\Gamma_n$ turns
$V_n$ into an orthogonal space whose isometry group
$W_n$ is generated by the mentioned transvections (3-transpositions)
$w_1,\ldots, w_n$; in this case, the relations
$(w_iw_j)^{m_{ij}}=1$ hold in
$W_n$ and, therefore, the mapping
$s_i\to w_i$ (
$i=1,\ldots,n$) is continued to the surjective homomorphism
$G_n\to W_n$. In the authors' previous paper, for all groups
$W_n=O^\pm_{2l}(2)$ (
$n=2l\geq 6$) and
$W_n= Sp_{2l}(2)$ (
$n=2l+1\geq 7$), 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
$O^\pm_{2l}(2)$ and
$Sp_{2l}(2)\times \mathbb{Z}_2$ (
$3\leq l\leq 10$) and the corresponding (infinite) Coxeter groups
$G_n$ with the difference in their genetic codes by exactly one gene (relation). For the groups
$W_n$ with the graphs
$\Gamma_n$ from the
$E$-series
$\{E_n\}$,
$\{ I_n\}$,
$\{ J_n\}$, and
$\{ K_n\}$, additional word relations are written explicitly.
Keywords:
groups with 3-transpositions, Coxeter graphs and groups, genetic codes.
UDC:
512. 544
MSC: 20C40 Received: 19.05.2020
Revised: 04.11.2020
Accepted: 16.11.2020
DOI:
10.21538/0134-4889-2020-26-4-234-243