A note on Weyl groups and root lattices
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 r...
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Veröffentlicht in: | Archiv der Mathematik 2018-11, Vol.111 (5), p.469-477 |
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Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
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Zusammenfassung: | 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
. |
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ISSN: | 0003-889X 1420-8938 |
DOI: | 10.1007/s00013-018-1234-5 |