A combinatorial derivation of the Poincare' polynomials of the finite irreducible Coxeter groups

(1) The Poincar e polynomials of the nite irreducible Coxeter groups and the Poincar e series of the aane Coxeter groups on three generators are derived by an elementary combinatorial method avoiding the use of Lie theory and invariant theory. (2) Non-recursive methods for the computation of`standard reduced words' for (signed) permutations are described. The algebraic basis for both (1) and (2) is a simple partition property of the weak Bruhat order of Coxeter groups into isomorphic parts. The combinatorial properties of weak Bruhat order of Coxeter groups, especially of the nite irreducible and aane ones, have been investigated for quite a time (see for example BW, St] and the references therein). But the partitioning property of the weak Bruhat order of Coxeter groups into isomorphic parts as stated in Theorem 0.1 below | though probably known by the experts | has certainly not been fully exploited. In the present paper we show the usefulness of this partition property by (1) giving a pictorial combinatorial derivation of the Poincar e polynomials and series for the nite irreducible Coxeter groups and the aane Coxeter groups on three generators | results, which until now have been obtained by invariant theoretic or Lie theoretic methods (cf. B, Hu]) | and by (2) deriving simple non-recursive schemes for the computation of standard reduced words for both unsigned and signed permutations. Some of the simplest pictures of the labeled Hasse diagrams in Section 1 have appeared also in connection with Verma modules and Schubert cells GM] and as a graphical device for calculating the homology of the most elementary Artin groups S]. Below in the introduction we recall some well known facts about Coxeter groups, weak Bruhat order, and Poincar e series, and | since we are not aware of any citable source | prove the basic partitioning theorem about the weak Bruhat order of Coxeter groups into order-isomorphic parts. Sections 1 and 3 contain pictures of the labeled Hasse diagrams for those parts of nite Coxeter groups and aane Coxeter groups on three generators, which are induced by the maximal parabolic subgroups with connected sub-Coxeter graphs. These pictures are derived directly from the group relations, and elementary calculations than yield the Poincar e polynomials or series for each of these Coxeter groups.