Factorizations in Schubert cells

For any reduced decomposition i = (i(1), i(2), ..., i(N)) of a permutation w and any ring R we construct a bijection P-i: (x(1), x(2), ..., x(N)) --> P-i1(x(1)) P-i2(x(2))... P-iN(x(N)) From R-N to the Schubert cell of w, whe re P-i1: (x(1)), P-i2(x(2)), ..., Pi(N)(x(N)) Stand for certain elementary matrices satisfying Coxeter-type relations. We show how to factor explicitl y any element of a Schubert cell into a product of such matrices. We apply this to give a one-to-one correspondence between the reduced decompositions of w and the injective balanced labellings of the diagram of w, and to cha racterize commutation classes of reduced decompositions. (C) 2000 Academic Press.