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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 $w, w' \in N$?

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up vote 4 down vote accepted

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.

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So is there an elementary proof for $SL(n,F)$, say using an algorithm how to transform the matrices instead of using a building? I was hoping for some argument $b w b' \sigma b''$ that $b'=b_1 b_2$ with $b_1^w, b_2^\sigma \in B$, or the like... – Marc Palm Jan 5 '12 at 16:37

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