Schubert cells and Whittaker functionals for GL(r,R) part I: Combinatorics

We give a formula for a birational map on the Schubert cell associated to each Weyl group element of G=GL(r) G = GL ( r ) . The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition of the flag variety G / B , where B is a Borel subgroup. An application of the formula includes a new proof of the existence of Whittaker functionals for principal series representations of GL(r,R) GL ( r , R ) via integration by parts. In this paper, we establish combinatorial properties of the birational map and prove auxiliary results.