Due to the comments, I have the impression that I have to be more precise.
Consider $G= GL_n(F)$ for a non-Archimedean field $F$ with ring of integers $o$. Let $K= GL_n(o)$ and let $I$ the Iwahori subroup of $K$ enlarged by the centrum. Let $M$ be the Levisubgroup (=diagonal matrices) of the standard Borel (=upper triangular matrices) in $G$ and $N$ the Normalizer (diagonal matrices * permutations) of $M$ in $G$.
Then $(K,M)$ and $(I,N)$ are is a BN pairs pair for $G$.
The Cartan decomposition is given as $$ G = K M K,$$ and the affine Bruhat decomposition as $$ G = INI.$$
Is it easier to read off the Cartan or alternatively the affine Bruhat decomposition, if the matrix is an upper triangular matrix? With read off, I mean is there a way to see the decomposition from the norms of the entries.
The usual algorithmic proof of the Cartan decomposition in the non-Archimedean case goes as follows:
Multiply by permutation matrices from the right and left such that we have at position $(1,1)$ the entry with maximal norm.
Kill the entries in the first column and first row by elementary matrices from $GL(n,o)$.
iterate the procedure for the matrix with the first column and row removed...
What is an algorithmic proof of the affine Bruhat decomposition, which works on $GL_n(F)$ rather than one the building?
