A note on Weyl groups and root lattices

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 t1, . . . tn ∈ T are reflections in W that generate W then P := t1, . . . tn−1 is a parabolic subgroup of (W, S) of rank n − 1 [BGRW17, 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 .

Interval groups related to finite Coxeter groups I

“…Notice also that the quasi-Coxeter elements in simply laced Coxeter groups are precisely those elements that admit a reduced decomposition into reflections such that the roots related to these reflections form a basis of the related root lattice (see [4]). In the non-simply laced case, then it is also required that the system of coroots generates the coroot lattice [4].…”

“…We continue to follow the cycle (2,4,3). Here the first two entries are 2 and 4 with 2 is not overlined.…”

“…Here the first two entries are 2 and 4 with 2 is not overlined. Then, we multiply w 2 from the right by the transposition (2,4). We therefore obtain a coefficient equal to 1 in diagonal position [2,2].…”

“…Now, we arrive at the last step. We multiply w 3 from the right by the transposition (3,4) and finally obtain the identity matrix denoted by…”

“…Consider the element w 0 corresponding to Equation 6. We have w 0 = w(1, 2) = (1)(2) (5,4,3). Let us write w 0 as a monomial matrix:…”