Algorithm for the cell multiplication rule for GL(n,F) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:30:38Z http://mathoverflow.net/feeds/question/84948 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/84948/algorithm-for-the-cell-multiplication-rule-for-gln-f Algorithm for the cell multiplication rule for GL(n,F) Marc Palm 2012-01-05T11:23:19Z 2012-01-05T16:54:33Z <p>Consider $F$ a non archimedean field and let $o$ be its ring of integer</p> <p>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).</p> <p>$B$ 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 $w, w' \in N$?</p> http://mathoverflow.net/questions/84948/algorithm-for-the-cell-multiplication-rule-for-gln-f/84961#84961 Answer by paul garrett for Algorithm for the cell multiplication rule for GL(n,F) paul garrett 2012-01-05T13:47:03Z 2012-01-05T16:54:33Z <p>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). </p> <p>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 <em>generating reflection</em> $\sigma$, namely: this is $Bw\sigma B$ when $\ell(w\sigma)>\ell(w)$, and is $Bw\sigma B\cup BwB$ for $\ell(w\sigma)&lt;\ell(\sigma)$. These are "axioms" for a BN-pair, but are <em>provable</em> from the action of SL(n) on the affine building of homothety classes of $o$-lattices.</p> <p>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.</p>