[...]... symplectic groups Sp(n) (sometimes denoted Sp(2n)), as well as the spherical buildings for other isometry groups with the sole exception of certain orthogonal groups O(n n) As in the case of GL(n + 1) and An , study of these buildings will yield an indirect proof that Cn really is the signed permutation group The ori amme Coxeter system Dn has generators which we write as with data and and s1 s2 s3... certain orthogonal groups on evendimensional vectorspaces over p-adic elds In all these cases, the `B' in the BN-pair is a compact open subgroup, called a Iwahori subgroup This will be explained in detail later when a ne buildings and Coxeter systems are de ned and examined carefully Garrett: `3 Chamber Complexes' 27 3 Chamber Complexes Chamber complexes The uniqueness lemma Foldings, half-apartments, walls,... sn;2 ) 3 = m(sn;2 sn ) = m(sn;2 s0n ) 2 = m(sn s0n ) (that is, they commute) and all other pairs commute Thus, unlike An and Cn , the element sn;2 has non-trivial relations with three other generators, and concommitantly the Coxeter diagram has a branch This system occurs in the spherical buildings for orthogonal groups on even-dimensional vectorspaces over algebraically closed elds, for example In this... one is meant to imagine that a zero-simplex is a point, a one-simplex is a line, a two-simplex is a triangle, a three simplex is a solid tetrahedron, and so on, and then that these things are stuck together along their faces in a reasonable sort of way to make up a simplicial complex Let V be a set, and X a set of nite subsets of V , with the property that, if x 2 X and if y x then y 2 X We also posit... 0 , the set Sv of elements occuring in any reduced expression for w `can be' a subset of T and `can be' a subset of T 0, so, by the corollary on subexpressions, Sv is a subset of T \ T 0 Thus, w 2 WT \T 0 , as desired 22 Garrett: 1 Coxeter Groups 0 Now let T and T 0 be distinct subsets of S and show that WT and WT are distinct By the previous assertion proven, we need only consider the case that T... is, commute) 3 = m(s1 s3 ) = m(s01 s3 ) = m(s3 s4 ) = : : : = m(sn;1 sn ) and 4 = m(sn sn+1 ) Thus, at the low-index end there is a branching, while at the high-index end there is a 4 appearing in the data This a ne single ori amme system occurs in the a ne building for orthogonal groups on odd-dimensional vectorspaces over p-adic elds, for example ~ The last in nite a ne family is Dn with n 4 This... `(w) = `(xs) + `T ((xs);1 w) and that xs 2 X , contradicting the assumed minimal length of x among elements of X Thus, we conclude that `(xs) > `(x) By induction on `(w), xes > 0 Similarly, we conclude that `(xt) > `(x) and xet > 0 8 Garrett: 1 Coxeter Groups It remains to show that yes > 0, e g , to show that yes = aes + bet with a b 0, since then wes = (xy)es = x(yes ) and we already know how x acts... roots, then ; = s ( ) = s ( ) = ; 2h i which implies that = h i Since both are unit vectors and are in + , we must have equality | are roots and = w for some w 2 W , then ws w;1 = s (The proof is direct computation, using the W -invariance of h i) | Lemma: If Proposition: For w 2 W and 2 w > 0 +, `(ws ) > `(w) if and only if Proof: It su ces to prove the `only if', since we can also consider the statement... that the group of symmetries of the data (or of the diagram) is just of order 2, the non-trivial symmetry being reversal of the indexing This ~ is much smaller than the symmetry group for An 26 Garrett: `2 Seven in nite families' ~ The system Cn appears in the a ne building for Sp(n) and unitary groups over a p-adic eld The corresponding spherical building at in nity, as described in the last chapter... re ections, and not just in S , makes the `left' and `right' de nitions equivalent If, by contrast, we give the analogous de nition with not T but S , then the distinction between vt and tv becomes signi cant The latter ordering is sometimes called a weak Bruhat order Proposition: Let v w and take s 2 S Then either vs w or vs ws or both Proof: First consider v ! w with vt = w for t 2 T and `(v) < `(w)