TY - JOUR
T1 - Coxeter covers of the classical coxeter groups
AU - Amram, Meirav
AU - Shwartz, Robert
AU - Teicher, Mina
N1 - Funding Information:
The first author was partially supported by the Emmy Noether Research Institute for Mathematics (center of the Minerva Foundation of Germany), the Excellency Center “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).
PY - 2010/12
Y1 - 2010/12
N2 - Let C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY(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 At,n be a group which contains t Abelian groups generated by n elements. The main result in this paper is that CY(T) is isomorphic to At,n ⋊ Bn or At,n ⋊ Dn, 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.
AB - Let C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either Bn or Dn. Let CY(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 At,n be a group which contains t Abelian groups generated by n elements. The main result in this paper is that CY(T) is isomorphic to At,n ⋊ Bn or At,n ⋊ Dn, 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.
KW - Classical Coxeter groups
KW - affine Coxeter groups
KW - signed graphs
KW - signed permutations
UR - http://www.scopus.com/inward/record.url?scp=78650744458&partnerID=8YFLogxK
U2 - 10.1142/S0218196710006023
DO - 10.1142/S0218196710006023
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AN - SCOPUS:78650744458
SN - 0218-1967
VL - 20
SP - 1041
EP - 1062
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
IS - 8
ER -