Factorizations in Schubert Cells

Abstract

For any reduced decomposition i=(i1, i2, ..., iN) of a permutation w and any ring R we construct a bijection Pi:(x1, x2, ..., xN)→Pi1(x1)Pi2(x2)...PiNN(xN) from RN 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. Copyright 2000 Academic Press.