An LUP algorithm for Lie group representations
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 L = KH, U = PMP-1, P is then an LUP
decomposition of A. The decomposition A=KNM is the so-called
Bruhat decomposition of an element of the general linear group. This
decomposition is well known in Lie group theory.
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.