On the connection of some groups generated by 3-transpositions with Coxeter groups

V. M. Sinitsin ; A. I. Sozutov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 234-243

Résumé

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 $m_{ii}=1$, $1\leq i$, 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.

Détail

Keywords: groups with 3-transpositions, Coxeter graphs and groups, genetic codes.