ON A PARABOLIC SYMMETRY OF FINITE COXETER GROUPS

Abstract

Let (W, S be a finite Coxeter system, and let J⊆S. Any w∈W has a unique factorization w = wJ wJ, where wj belongs to the parabolic subgroup WJ generated by J, and wJ is of minimal length in the coset wWJ. It is shown here that wI = wJ if and only if wI = wJ, for all I, J ⊆ S. Furthermore, a similar symmetry property in arbitrary (WI, WJ-double cosets is conjectured, which links this result to the Solomon descent algebra of W. 2000 Mathematics Subject Classification 20F55.