Coxeter covers of the classical coxeter groups

Meirav Amram; Robert Shwartz; Mina Teicher

doi:10.1142/S0218196710006023

Coxeter covers of the classical coxeter groups

Meirav Amram
, Robert Shwartz
, Mina Teicher

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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 ⋊ B_n or A_t,n ⋊ 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.

Original language	English
Pages (from-to)	1041-1062
Number of pages	22
Journal	International Journal of Algebra and Computation
Volume	20
Issue number	8
DOIs	https://doi.org/10.1142/S0218196710006023
State	Published - Dec 2010
Externally published	Yes

Keywords

Classical Coxeter groups
affine Coxeter groups
signed graphs
signed permutations

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10.1142/S0218196710006023

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title = "Coxeter covers of the classical coxeter groups",

abstract = "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.",

keywords = "Classical Coxeter groups, affine Coxeter groups, signed graphs, signed permutations",

author = "Meirav Amram and Robert Shwartz and Mina Teicher",

note = "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 {\textquotedblleft}Group Theoretic Methods in the Study of Algebraic Varieties{\textquotedblright} of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).",

year = "2010",

month = dec,

doi = "10.1142/S0218196710006023",

language = "אנגלית",

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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

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EP - 1062

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

IS - 8

ER -

Coxeter covers of the classical coxeter groups

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Keywords

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