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.