The reverse decomposition of unipotents for bivectors

TL;DR

This paper develops a method to express elementary generators as products of conjugates of matrices and their inverses in the context of the second fundamental representation of the general linear group, with stabilization results for certain matrix columns.

Contribution

It introduces uniform polynomial expressions for elementary generators in the second fundamental representation, advancing understanding of matrix decompositions in algebraic groups.

Findings

Polynomial expressions of elementary generators are constructed.

Stabilization theorems for columns of specific matrix groups are proved.

Results apply to matrices over commutative rings and exterior square representations.

Abstract

For the second fundamental representation of the general linear group over a commutative ring $R$ we construct straightforward and uniform polynomial expressions of elementary generators as products of elementary conjugates of an arbitrary matrix and its inverse. Towards the solution we get stabilization theorems for any column of a matrix from $GL_{n \choose 2}(R)$ or from the exterior square of $GL_n(R)$, $n\geq 3$.