Schubert cells and Whittaker functionals for $\text{GL}(n,\mathbb{R})$ part I: Combinatorics

TL;DR

This paper introduces a combinatorial formula for a birational map on Schubert cells in $ ext{GL}(n, ext{R})$, simplifying matrix decompositions and providing new insights into the structure of the flag variety and Whittaker functionals.

Contribution

It presents a novel combinatorial formula for a birational map on Schubert cells, aiding in understanding matrix decompositions and Whittaker functional existence.

Findings

Derived a formula for the birational map on Schubert cells.

Proved combinatorial properties of the birational map.

Provided a new proof of Whittaker functional existence.

Abstract

We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $G=\text{GL}(n)$. 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 $\text{GL}(n,\mathbb{R})$ via integration by parts. In this paper, we establish combinatorial properties of the birational map and prove auxiliary results.