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

Department of Mathematics

Bar-Ilan University

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

Bibliographical 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 “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).

Funding

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).

Funders	Funder number
Emmy Noether Research Institute for Mathematics
Israel Science Foundation	HPRN-CT-2009-00099

Keywords

Classical Coxeter groups
affine Coxeter groups
signed graphs
signed permutations

Access to Document

10.1142/S0218196710006023

Cite this

@article{c53ab28661714c7b9b4a049b6292d383,

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 “Group Theoretic Methods in the Study of Algebraic Varieties” of the Israel Science Foundation, and EAGER (EU network, HPRN-CT-2009-00099).",

year = "2010",

month = dec,

doi = "10.1142/S0218196710006023",

language = "אנגלית",

volume = "20",

pages = "1041--1062",

journal = "International Journal of Algebra and Computation",

issn = "0218-1967",

publisher = "World Scientific Publishing Co. Pte Ltd",

number = "8",

}

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

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

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 -

Coxeter covers of the classical coxeter groups

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this