Coxeter covers of the classical Coxeter groups

2008/03/20 by Meirav Amram, M. Amram, R. Shwartz +6
Computer Science · Mathematics · #20B30 #20E34 #20F05 #20F55 #20F65 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #math.AG #math.GR #msc:20B30 #msc:20E34 #msc:20F05 #msc:20F55 #msc:20F65 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0803.3010

26 pages, 7 figures

arxiv created 2008/03/20 · openalex publication_date 2008/03/20 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either B_n or D_n. Let C_Y(T) be a natural quotient of C(T), and if C(T) is simply-laced (which means all the relations between the generators has order 2 or 3), C_Y(T) is a generalized Coxeter group, too . Let A_t,n be a group which contains t Abelian groups generated by n elements. The main result in this paper is that C_Y(T) is isomorphic to A_t,n \semidirect B_n or A_t,n \semidirect D_n, depends on whether the signed graph T contains loops or not, or in other words C(T) is simply-laced or not, and t is the number of the cycles in T. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.

Related

• Coxeter covers of the symmetric groups
• Interval groups related to finite Coxeter groups I
• Chordal Coxeter Groups
• Simply-laced Coxeter groups and groups generated by symplectic transvections
• Simply-laced mixed-sign Coxeter groups, with an associate graph is a line or a simple cycle