# Geometric interpretation of $BN$-pairs

My question is relative to a geometric interpretation of the $BN$-pairs that arise in Tits' theory of buildings. Here is a definition that comes from an article by G. Stroth (Nonspherical spheres).

$[\ldots]$

Let $\mathcal{P} = \{P_1, \ldots, P_n\}$ be a minimal parabolic system for a group $G$, $B=P_1 \cap \ldots \cap P_n$ the Borel subgroup. A subgroup $N$ of $G$ is called a Weyl group for $\mathcal{P}$ iff

1) $N= \langle x_1, \ldots, x_n \rangle, x_i \in P_i-B, x_i^2 \in B$.

2) $B \cap N$ is a normal subgroup of $N$.

3) $N \cap P_i = (B \cap N) \langle x_i \rangle , i = 1, \ldots, n$.

$[\ldots]$

(A) $G = BNB$ and

(B) $BgBhB \subset (BgB) \cup (BghB)$ for all $g,h \in N$

then we have a $BN$-pair. Geometrically, the Weyl group $N$ is the stabilizer of an apartment $\Delta$ of the geometry $\Gamma$ defined from $\mathcal{P}$, and $B$ is the stabilizer of a chamber of $\Delta$. I am trying to get a clear geometric view of these objects.

Here is my question. What could be a geometric interpretation of condition (B)?

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I think you must have copied down (B) wrong, since none of the things that are supposed to be examples satisy it. Probably you want $h$ to be one of the $x_i$'s. –  Ben Webster Feb 20 '11 at 17:11
@Ben Webster: I checked again, and I made no mistake in copying it down. –  Thomas Connor Feb 20 '11 at 17:31
Fair enough. That definition is stronger than the usual definition of B,N-pair (en.wikipedia.org/wiki/(B,_N)_pair) and rules out many interesting examples. –  Ben Webster Feb 20 '11 at 17:46
That condition (B), whether copied accurately or not, is troubling. For example, in GL(3,k) over a field k, the largest (spherical) Bruhat cell $Bw_oB$ (with longest Weyl element $w_o$) is such that $Bw_oB\cdot Bw_oB=GL(3,k)$. That is, all 3! of the Bruhat cells are hit. The condition (B) above would require that $Bw_oB\cdot Bw_oB=B\cup Bw_oB$. As Ben W. noted above, it seems likely that $h$ in (B) should be among the $x_i$'s. –  paul garrett Jun 27 '11 at 20:46

Note: I'm using the general definition of BN-pair, which is weaker than the condition (B) given above.

It's a triangle inequality. One way to think about BN-pair is that they give a sort of combinatorial distance function on $G/B$. Given two cosets $g_1B$ and $g_2B$, you look at the product $Bg_1^{-1}g_2 B\in B\backslash G/B\cong N/(N\cap B):= W$ and think of that as the "distance" between them. To get a more numberish distance, you can let the length of an element of $W$ be the length of the shortest product of $x_i$'s which gives it.

If I take two cosets $BgB$ and $BhB$, and expand those as $Bx_{i_1}\cdots x_{i_n}B$ and $Bx_{j_1}\cdots x_{j_m}B$, then $$BgB\cdot BhB\subset Bx_{i_1}\cdots x_{i_n}B\cdot Bx_{j_1}\cdots x_{j_m}B\subset Bx_{i_1}B\cdots Bx_{i_n}Bx_{j_1}B\cdots Bx_{j_m}B.$$ Applying (B) inductively, we see that the last term is in the union of certain double cosets which have length shorter than the sum of that of $BgB$ and $BhB$.

To apply this to the "distance function," note that $Bg_1^{-1}g_3B\subset Bg_1^{-1}g_2B\cdot Bg_2^{-1}g_3B$, so the length of the distance between $g_1$ and $g_3$ is less than the sum of the lengths for $g_1$ to $g_2$ and $g_2$ to $g_3$: the triangle inequality.

Of course, that just shows that (B) implies the triangle inequality, but it's easy to see that it's also a special case.

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