## Details

Original language | English |
---|---|

Pages (from-to) | 469-477 |

Number of pages | 9 |

Journal | Archiv der Mathematik |

Volume | 111 |

Issue number | 5 |

Early online date | 14 Sept 2018 |

Publication status | Published - Nov 2018 |

Externally published | Yes |

## Abstract

We follow the dual approach to Coxeter systems and show for Weyl groups that a set of reflections generates the group if and only if the related sets of roots and coroots generate the root and the coroot lattices, respectively. Previously, we have proven if (W, S) is a Coxeter system of finite rank n with set of reflections T and if t
_{1}, … t
_{n}∈ T are reflections in W that generate W, then P: = ⟨ t
_{1}, … t
_{n}
_{-}
_{1}⟩ is a parabolic subgroup of (W, S) of rank n- 1 (Baumeister et al. in J Group Theory 20:103–131, 2017, Theorem 1.5). Here we show if (W, S) is crystallographic as well, then all the reflections t∈ T such that ⟨ P, t⟩ = W form a single orbit under conjugation by P.

## Keywords

- Dual Coxeter system, Finite crystallographic Coxeter systems, Finite crystallographic root lattices, Generation of dual Coxeter systems, Quasi-Coxeter elements, Weyl group

## ASJC Scopus subject areas

## Cite this

- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS

**A note on Weyl groups and root lattices.**/ Wegener, Patrick; Baumeister, Barbara.

In: Archiv der Mathematik, Vol. 111, No. 5, 11.2018, p. 469-477.

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

*Archiv der Mathematik*, vol. 111, no. 5, pp. 469-477. https://doi.org/10.1007/s00013-018-1234-5

*Archiv der Mathematik*,

*111*(5), 469-477. https://doi.org/10.1007/s00013-018-1234-5

}

TY - JOUR

T1 - A note on Weyl groups and root lattices

AU - Wegener, Patrick

AU - Baumeister, Barbara

N1 - Publisher Copyright: © 2018, Springer Nature Switzerland AG.

PY - 2018/11

Y1 - 2018/11

N2 - We follow the dual approach to Coxeter systems and show for Weyl groups that a set of reflections generates the group if and only if the related sets of roots and coroots generate the root and the coroot lattices, respectively. Previously, we have proven if (W, S) is a Coxeter system of finite rank n with set of reflections T and if t 1, … t n∈ T are reflections in W that generate W, then P: = ⟨ t 1, … t n - 1⟩ is a parabolic subgroup of (W, S) of rank n- 1 (Baumeister et al. in J Group Theory 20:103–131, 2017, Theorem 1.5). Here we show if (W, S) is crystallographic as well, then all the reflections t∈ T such that ⟨ P, t⟩ = W form a single orbit under conjugation by P.

AB - We follow the dual approach to Coxeter systems and show for Weyl groups that a set of reflections generates the group if and only if the related sets of roots and coroots generate the root and the coroot lattices, respectively. Previously, we have proven if (W, S) is a Coxeter system of finite rank n with set of reflections T and if t 1, … t n∈ T are reflections in W that generate W, then P: = ⟨ t 1, … t n - 1⟩ is a parabolic subgroup of (W, S) of rank n- 1 (Baumeister et al. in J Group Theory 20:103–131, 2017, Theorem 1.5). Here we show if (W, S) is crystallographic as well, then all the reflections t∈ T such that ⟨ P, t⟩ = W form a single orbit under conjugation by P.

KW - Dual Coxeter system

KW - Finite crystallographic Coxeter systems

KW - Finite crystallographic root lattices

KW - Generation of dual Coxeter systems

KW - Quasi-Coxeter elements

KW - Weyl group

UR - http://www.scopus.com/inward/record.url?scp=85053527229&partnerID=8YFLogxK

U2 - 10.1007/s00013-018-1234-5

DO - 10.1007/s00013-018-1234-5

M3 - Article

VL - 111

SP - 469

EP - 477

JO - Archiv der Mathematik

JF - Archiv der Mathematik

SN - 0003-889X

IS - 5

ER -