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.

Bibliographie

[1] Fischer B., Finite groups generated by 3-transpositions, WMI Preprints, University of Warwick, Coventry (UK), 1969

[2] Gorenstein D., Konechnye prostye gruppy, Mir, M., 1985, 352 pp.

[3] McLaughlin J., “Some subgroups of $SL_n(F_2)$”, Ill. J. Math., 13:1 (1969), 108–115

[4] Sozutov A.I., “O gruppakh tipa $\Sigma _4$, porozhdennykh 3-transpozitsiyami”, Sib. mat. zhurn., 33:1 (1992), 140–149

[5] Hall J.I., “Graphs, geometry, 3-transposition, and symplectic $F_2$-transvection groups”, Proc. London Math. Soc., 58 (1989), 89–111

[6] Sozutov A.I., “Ob algebrakh Li s monomialnym bazisom”, Sib. mat. zhurn., 34:5 (1993), 188–201

[7] Aschbacher M., 3-transposition groups, Cambridge University Press, Cambridge, 1997, 260 pp.

[8] Matsuo A., “3-transposition groups of symplectic type and vertex operator algebras”, J. Math. Soc. Japan, 57:3 (2005), 639–649

[9] Sozutov A.I., Kuznetsov A.A., Sinitsin V.M., “O sistemakh porozhdayuschikh nekotorykh grupp s 3-transpozitsiyami”, Sib. mat. elektron. izv., 10 (2013), 285–301 | DOI

[10] Hall J.I., Shpectorov S., “The spectra of finite 3-transpositions groups”, [e-resource], 2018, 35 pp., arXiv: http://arxiv.org/abs/1809.03696

[11] Griess R.L. Jr., “A vertex operator algebra related to $E_8$ with avtomorphism group $O^+(10,2)$”, Ohio State Univ. Math. Res. Inst., 7 (1998), 43–58, Publ. de Gruyter, Berlin

[12] Cuypers H., Horn M., J. in't panhuis., Shpectorov S., “Lie algebras and 3-transpositions”, [e-resource], 2011, 23 pp., arXiv: http://arxiv.org/abs/1104.0536

[13] Burbaki N., Gruppy i algebry Li. Gruppy, porozhdennye otrazheniyami. Gl. IV-VI, Mir, M., 1972, 334 pp.

[14] Kokseter G.S.M., Mozer U.O.Dzh., Porozhdayuschie elementy i opredelyayuschie elementy diskretnykh grupp, Nauka, M., 1980, 240 pp.

[15] Kondratev A.S., Gruppy i algebry Li, Izd-vo UrO RAN, Ekaterinburg, 2009, 310 pp.

[16] O'Mira O., Lektsii o simplekticheskikh gruppakh, Mir, M., 1979, 167 pp.

[17] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, An atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp.