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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:

1. Multiply by permutation matrices from the right and left such that we have at position $(1,1)$ the entry with maximal norm.

2. Kill the entries in the first column and first row by elementary matrices from $GL(n,o)$.

3. 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?

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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 $K$. K$enlarged by the centrum. Let$M$be the Levisubgroup (diagonal =diagonal matrices) of the standard Borel (=upper triangular matrices) in$G$and$N$the Normalizer (diagonal matrices * permutations) of$M$in$G$. Then$(K,M)$and$(I,N)$are BN pairs for$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 algorithm algorithmic proof of the Cartan decomposition in the non-Archimedean case goes as follows: 1. Multiply by permutation matrices from the right and left such that we have at position$(1,1)$the entry with maximal norm. 2. Kill the entries in the first column and first row by elementary matrices from$GL(n,o)$. 3. 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? 5 edited body 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$I$the Iwahori subroup of$K$. Let$M$be the Levisubgroup (diagonal matrices) of the standard Borel in$G$and$N$the Normalizer (diagonal matrices * permutations) of$M$in$G$. Then$(K,M)$and$(I,M)$(I,N)$ are BN pairs 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 algorithm of the Cartan decomposition in the non-Archimedean case goes as follows:

1. Multiply by permutation matrices from the right and left such that we have at position $(1,1)$ the entry with maximal norm.

2. Kill the entries in the first column and first row by elementary matrices from $GL(n,o)$.

3. iterate the procedure for the matrix with the first column and row removed...

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