Big cells and LU factorization in reductive monoids

Abstract

Abstract. It is well known that an invertible matrix admits a factorization as a product of a lower triangular matrix L and an upper triangular matrix U if and only if all the principal minors of the matrix are non-zero. The corresponding problem for singular matrices is much more subtle. We study this problem in the general setting of a reductive monoid and obtain a solution in terms of the Bruhat-Chevalley order. In the process we obtain a decomposition of the big cell B − B of a reductive monoid, where B and B − are opposite Borel subgroups of the unit group.