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
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In: Archiv der Mathematik, Vol. 111, No. 5, 11.2018, p. 469-477.
Research output: Contribution to journal › Article › Research › peer review
}
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 -