Arjeh M. Cohen

The classical LUP algorithm decomposes a matrix into a product of a
Lower triangular matrix *L*, an Upper triangular matrix *U* and a
Permutation matrix *P*. The matrix *U* can be chosen so that
*PUP ^{-1} * is lower triangular.

If the input matrix *A* is invertible, the decomposition can be found
by elementary row and column operations on *A*, thereby keeping the
column operations to a minimum. The result is then a triple *K*, *N*,
*M* of two lower triangular matrices *K* and *M* and a monomial matrix
*N* (that is, *N= HP*, the product of diagonal matrix *H* and a
permutation matrix *P*) such that *A = KNM* and *NMN ^{-1}* is upper
triangular. The triple

In this context the question arises whether in any linear (highest
weight) representation of a Lie group we can find the Bruhat
decomposition of a Lie group element given by a matrix in the
representation. In the talk we shall address this question and
interpret the above LUP decomposition as a special case.