Factorizations in Schubert cells

For any reduced decomposition i = (i1, i2, . . . , iN ) of a permutation w and any ring R we construct a bijection Pi : (x1, x2, . . . , xN ) 7→ Pi1(x1)Pi2(x2) · · · PiN (xN ) from R to the Schubert cell of w, where Pi1(x1), Pi2(x2), . . . , PiN (xN ) stand for certain elementary matrices satisfying Coxeter-type relations. We show how to factor explicitly 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 characterize commutation classes of reduced decompositions. Mathematics Subject Classification (1991): 20B30, 20G15, 05E15, 14M15, 15A23