Big cells and LU factorization in reductive monoids

Putcha, Mohan

doi:10.1090/S0002-9939-02-06515-2

Big cells and LU factorization in reductive monoids
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by Mohan S. Putcha PDF

Proc. Amer. Math. Soc. 130 (2002), 3507-3513 Request permission

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 $\overline B{}^-\overline B$ of a reductive monoid, where $B$ and $B^-$ are opposite Borel subgroups of the unit group.

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