Consider $F$ a non archimedean field and let $o$ be its ring of integer


Let $B$ be the Iwahori subgroup of $GL\_n(F)$ (resp. $GL\_n(o)$) and let $N$ be the normalizer of the diagonal matrices (respective the diagonal matrices).


$B$ and $N$ give a $BN$ pair for $GL\_n(F)$. Is there an explicit algorithm on the group level verifying the cell multiplication rule 
$$ B w B \cdot B w' B \subset B w w' B \amalg B w B$$
for $w, w' \in N$?



Algorithm for the cell multiplication rule for GL(n,F)
The asserted cell multiplication isn't quite right as it stands. First, GL(n) does not have "strict" BN-pair structure, but SL(n) does. An obvious extra element needs to be added for GL(n). 


Second, for the strict BN-pair situation of SL(n,F) and SL(n,o), the cell multiplication rules are all generated by two cases of $BwB\cdot B\sigma B$ for general element $w$ and *generating reflection* $\sigma$, namely: this is $Bw\sigma B$ when $\ell(w\sigma)>\ell(w)$, and is $Bw\sigma B\cup BwB$ for $\ell(w\sigma)<\ell(\sigma)$. These are "axioms" for a BN-pair, but are *provable* from the action of SL(n) on the affine building of homothety classes of $o$-lattices.


Edit: @pm, a genuinely implementable algorithm to produce the Bruhat decomposition $bwb'$ of given $g\in G$ (say, in a strict BN-pair) may be non-trivial (and I do not have any truly usefu idea about this off-hand), since, in any case, there is ambiguity in the $b,b'$ in the Iwahori, especially in the affine case. The arguments I know for cell multiplication only refer to double cosets $BwB$, and their "geometric" interpretation in terms of the building, not individuals.


