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Proposition 3.124. Let \( A \) and \( B \) be arbitrary simplices of \( \sum \left( {W, S}\right) \) . Choose a chamber \( C \geq {AB} \) . Then every minimal gallery from \( A \) to \( B \) is equivalent under \( {W}_{A} \cap {W}_{B} \) to one that starts with \( C \) . In particular, there are only finitely many \( \... | Proof. By the lemma and Exercise 3.13, \( {W}_{A} \cap {W}_{B} \) is transitive on \( {\mathcal{C}}_{AB} \) . This implies the first assertion. For the second assertion, we need only recall that a minimal gallery from \( A \) to \( B \) starting with \( C \) must end with \( {BC} \), so there are only finitely many of ... | No |
Proposition 3.125. Let \( \sum \) be an irreducible Coxeter complex, and let \( H \) be a wall of \( \sum \) . Then \( \sum \) contains a chamber \( C \) that is disjoint from \( H \), in the sense that none of the vertices of \( C \) are in \( H \) . More generally, every simplex \( A \) disjoint from \( H \) is a fac... | Proof of the proposition. Let \( A \) be disjoint from \( H \), and choose a maximal simplex \( B \geq A \) disjoint from \( H \) . Clearly \( B \) is not the empty simplex, since \( H \) cannot contain every vertex of \( \sum \) . We will show that \( B \) is a chamber. Let \( \tau \) be a type function on \( \sum \) ... | Yes |
Lemma 3.130. Let \( \sum \) be a Coxeter complex.\n\n(1) Every convex chamber subcomplex of \( \sum \) is a subsemigroup of \( \sum \) . | Proof. (1) Let \( {\sum }^{\prime } \) be a convex chamber subcomplex of \( \sum \) . Given \( A, B \in {\sum }^{\prime } \), we must show that \( {AB} \in {\sum }^{\prime } \) . Choose a chamber \( C \in {\sum }^{\prime } \) with \( C \geq B \) . By checking sign sequences, one sees that \( {AB} \leq {AC} \) ; so it s... | Yes |
Theorem 3.131. Let \( \sum \) be a Coxeter complex and \( {\sum }^{\prime } \) a subcomplex containing at least one chamber. Then the following conditions are equivalent:\n\n(i) \( {\sum }^{\prime } \) is a convex chamber subcomplex of \( \sum \) .\n\n(ii) \( {\sum }^{\prime } \) is an intersection of roots.\n\n(iii) \... | Proof. The equivalence of (i) and (ii) is almost immediate from Proposition 3.94, but one must be a little careful: Suppose (i) holds, and let \( {\sum }^{\prime \prime } \) be the intersection of all roots containing \( {\sum }^{\prime } \) . Then \( {\sum }^{\prime \prime } \) is a chamber subcom-plex of \( \sum \) b... | Yes |
Proposition 3.137. Let \( {\sum }^{\prime } \) be a subcomplex of a Coxeter complex \( \sum \) . Then the following conditions are equivalent:\n\n(i) \( {\sum }^{\prime } \) is convex.\n\n(ii) \( {\sum }^{\prime } \) is an intersection of roots.\n\n(iii) \( {\sum }^{\prime } \) is an intersection of convex chamber subc... | It is trivial that an intersection of convex subcomplexes is convex. So the essential content of Proposition 3.137 is the implication (i) \( \Rightarrow \) (ii). This has a concrete interpretation in terms of sign sequences. It says that every convex subcomplex is defined by conditions of the form \( {\sigma }_{H}\left... | No |
Lemma 3.140. Let \( {\sum }^{\prime } \) be a convex subcomplex of \( \sum \). (1) All maximal simplices of \( {\sum }^{\prime } \) have the same dimension. (2) If \( A \) is a maximal simplex of \( {\sum }^{\prime } \), then \( {\sum }^{\prime } \subseteq \operatorname{supp}A \). | Proof. (1) If \( A \) and \( B \) are maximal simplices of \( {\sum }^{\prime } \), then \( {AB} = A \) and \( {BA} = B \) , so \( A \) and \( B \) have the same dimension by Corollary 3.113. (2) For any simplex \( B \in {\sum }^{\prime } \), we have \( {AB} = A \) by maximality of \( A \) ; hence \( \operatorname{supp... | Yes |
Corollary 3.141.\n\n(1) For any simplex \( A \in \sum \), we have \( \dim A = \dim \left( {\operatorname{supp}A}\right) \).\n\n(2) Let \( A \) and \( B \) be simplices in \( \sum \) with \( \dim A = \dim B \). If \( B \in \operatorname{supp}A \), then \( \operatorname{supp}A = \operatorname{supp}B \). | Proof. (1) The subcomplex supp \( A \) is convex, so all of its maximal simplices have the same dimension by the lemma; now apply Proposition 3.99.\n\n(2) \( B \) is a maximal simplex of supp \( A \), so \( \operatorname{supp}A \subseteq \operatorname{supp}B \) by the lemma. The opposite inclusion is immediate from the... | Yes |
Lemma 3.142. If \( {\sum }^{\prime } \) is a convex subcomplex of \( \sum \), then any two maximal simplices \( A, B \in {\sum }^{\prime } \) can be connected by a \( {\sum }^{\prime } \) -gallery. | Proof. We argue by induction on \( d\left( {A, B}\right) \), where the latter denotes the gallery distance between \( A \) and \( B \) in \( \sum \) (see Section A.1.3). If \( d\left( {A, B}\right) = 0 \), then \( A \) and \( B \) are joinable in \( \sum \), and one sees immediately by using sign sequences that \( {AB}... | Yes |
Lemma 3.143. Let \( x \) be a vertex of \( \sum \) . Then \( \operatorname{supp}x \) is 0 -dimensional and either has \( x \) as its only vertex or else has exactly two vertices \( x, y \) . | The combinatorial proof of this is somewhat tricky. To avoid disrupting the flow of ideas, we postpone the proof to the next subsection. | No |
Lemma 3.159. Let \( {\sum }^{\prime } \) be a 1-dimensional convex subcomplex of \( \sum \), and let \( x \) and \( y \) be two distinct vertices of \( {\sum }^{\prime } \). For any edge \( F \) of \( {\sum }^{\prime } \) such that \( {yx} \leq F \), there is a sequence \( x = {x}_{0},\ldots ,{x}_{n} = y \) of vertices... | Proof. Note that the construction of the \( {x}_{i} \) is forced on us by properties (1)-(5): We must have \( {E}_{1} = {xF} \) by (3), and then \( {x}_{1} \) must be the vertex of \( {E}_{1} \) different from \( x = {x}_{0} \). If \( {x}_{1} = y \), we are done; otherwise, we must have \( {E}_{2} = {x}_{1}F \), and so... | Yes |
Lemma 3.162. Let \( A \) and \( {\sum }^{\prime } \) be as above, let \( \alpha \) be a root of \( \sum \) with \( A \in \partial \alpha \) , and let \( {\alpha }^{\prime } \mathrel{\text{:=}} \alpha \cap {\sum }^{\prime } \) . Then\n\n\[ \n\mathcal{C}\left( \alpha \right) = \left\{ {C \in \mathcal{C}\left( \sum \right... | Proof. Let \( C \) be a chamber of \( \sum \) . Using the convexity of roots and Remark 3.80 , we have\n\n\[ \nC \in \alpha \Rightarrow {AC} \in \alpha \cap {\sum }_{ \geq A} \Rightarrow \left( {{AC} \smallsetminus A}\right) \in {\alpha }^{\prime }.\n\]\n\nSimilarly,\n\n\[ \nC \in - \alpha \Rightarrow {AC} \in - \alpha... | Yes |
Lemma 3.164. Let \( \alpha \) and \( \beta \) be roots of \( \sum \) with \( \alpha \neq \pm \beta \) . If the pairs \( \{ \alpha ,\beta \} \) and \( \{ - \alpha ,\beta \} \) are both non-nested, then \( \partial \alpha \cap \partial \beta \) is a chamber complex of codi-mension 2 in \( \sum \), and \( {\operatorname{l... | Proof. The hypothesis says that the intersection \( \left( {\pm \alpha }\right) \cap \left( {\pm \beta }\right) \) contains a chamber for each of the four possible choices of sign. Considering a minimal gallery from a chamber in \( \alpha \cap \beta \) to a chamber in \( \alpha \cap - \beta \), we obtain a panel \( P \... | Yes |
Proposition 3.165. Let \( \alpha \) and \( \beta \) be roots of \( \sum \) with \( \alpha \neq \pm \beta \) . Then the following conditions are equivalent:\n\n(i) Either \( \{ \alpha ,\beta \} \) is nested or \( \{ - \alpha ,\beta \} \) is nested.\n\n(ii) The product \( {s}_{\alpha }{s}_{\beta } \) has infinite order.\... | Proof. (i) \( \Rightarrow \) (ii): Suppose one of the pairs is nested, say \( \alpha \subsetneqq \beta \) . Setting \( w \mathrel{\text{:=}} {s}_{\alpha }{s}_{\beta } \), we then have\n\n\[ \n{w\alpha } \subsetneqq {w\beta } = {s}_{\alpha }\left( {-\beta }\right) \subsetneqq {s}_{\alpha }\left( {-\alpha }\right) = \alp... | Yes |
Corollary 3.166. \( \sum \) has nested roots if and only if it is not spherical. | Proof. We already know that spherical Coxeter complexes do not have nested roots (Lemma 3.53). If \( \sum \) is not spherical, on the other hand, then it has a pair of reflections whose product has infinite order by Proposition 2.74, so it has a pair of nested roots by Proposition 3.165. | Yes |
Corollary 3.167. If \( \left( {W, S}\right) \) is a Coxeter system and \( {W}^{\prime } \leq W \) is a finite subgroup generated by two reflections, then \( {W}^{\prime } \) is contained in a finite parabolic subgroup of \( W \) of rank 2. | Proof. Let \( \sum \mathrel{\text{:=}} \sum \left( {W, S}\right) \) . Combining the proposition with Lemma 3.164, we see that \( {W}^{\prime } \) stabilizes a codimension-2 simplex with spherical link. The stabilizer of this link is a finite parabolic subgroup of \( W \) of rank 2. | No |
Proposition 4.6. \( \Delta \) is colorable. Moreover, the isomorphisms \( \sum \rightarrow {\sum }^{\prime } \) in axiom (B2) can be taken to be type-preserving. | Proof. Fix an arbitrary chamber \( C \), and assign types to its vertices arbitrarily. If \( \sum \) is any apartment containing \( C \), then [since Coxeter complexes are colorable] the assignment of types on \( C \) extends uniquely to a type function \( {\tau }_{\sum } \) of \( \sum \) . For any two such apartments ... | Yes |
Proposition 4.9. If \( \Delta \) is a building, then so is \( \operatorname{lk}A \) for any \( A \in \Delta \) . In particular, the link is a chamber complex. | Proof. Choose a fixed system of apartments \( \mathcal{A} \) for \( \Delta \) . Given \( A \in \Delta \), let \( {\mathcal{A}}^{\prime } \) be the family of subcomplexes of \( {\operatorname{lk}}_{\Delta }A \) of the form \( {\operatorname{lk}}_{\sum }A \), where \( \sum \) is an element of \( \mathcal{A} \) containing... | Yes |
Suppose \( \Delta \) is a building of rank 1 (dimension 0). Then every apartment must be a 0 -sphere \( {S}^{0} \), since this is the only Coxeter complex of rank 1. | In particular, \( \Delta \) must have at least two vertices. Conversely, a rank-1 simplicial complex with at least two vertices is a building (with every 2-vertex subcomplex as an apartment). Thus the rank-1 buildings are precisely the flag complexes of the rank-1 incidence geometries with at least 2 points. [It is, of... | No |
Suppose \( \Delta \) is a building of rank 2 (dimension 1). Then an apartment \( \sum \) must be a \( {2m} \) -gon for some \( m\left( {2 \leq m \leq \infty }\right) \) . | Let’s begin with the case \( m = 2 \) . Then every apartment is a quadrilateral:\n\n\n\n(As usual, the two colors, black and white, represent the two types of vertices.) It follows easily from the building axioms tha... | No |
Proposition 4.33. Every apartment is a retract of \( \Delta \) . | Proof. This is very similar to the construction of a type function. Fix a chamber \( C \) of the given apartment \( \sum \), and consider all the apartments \( {\sum }^{\prime } \) that contain \( C \) . For any such \( {\sum }^{\prime } \) there is a unique isomorphism \( {\phi }_{{\sum }^{\prime }} : {\sum }^{\prime ... | Yes |
Corollary 4.34. Let \( C \) and \( D \) be chambers of \( \Delta \), and let \( \sum \) be any apartment containing \( C \) and \( D \) . Then \( {d}_{\Delta }\left( {C, D}\right) = {d}_{\sum }\left( {C, D}\right) \) . Consequently, the diameter of \( \Delta \) is equal to the diameter of any apartment. | Proof. Suppose \( \Gamma \) is a minimal gallery in \( \sum \) from \( C \) to \( D \) . Then \( \Gamma \) is also minimal in \( \Delta \) ; for if there were a shorter gallery in \( \Delta \), then we could get a shorter one in \( \sum \) by applying a retraction. This proves the first assertion. As an immediate conse... | Yes |
Corollary 4.36. The Coxeter matrix \( M \) depends only on \( \Delta \), not on the system of apartments. It is given by\n\n\[ m\left( {s, t}\right) = \operatorname{diam}\left( {{\operatorname{lk}}_{\Delta }A}\right) \]\n\nwhere \( A \) is any simplex of cotype \( \{ s, t\} \) . | Proof. Suppose \( A \) has cotype \( \{ s, t\} \), and let \( \sum \) be any apartment containing \( A \) . Then we have \( m\left( {s, t}\right) = \operatorname{diam}\left( {{\operatorname{lk}}_{\sum }A}\right) \) by definition. But \( {\operatorname{lk}}_{\sum }A \) is an apartment in the building \( {\operatorname{l... | Yes |
Proposition 4.39. The retraction \( \rho = {\rho }_{\sum, C} \) has the following properties.\n\n(1) For any face \( A \leq C,{\rho }^{-1}\left( A\right) = \{ A\} \) .\n\n\( \left( 2\right) \rho \) preserves distances from \( C \), i.e., \( d\left( {C,\rho \left( D\right) }\right) = d\left( {C, D}\right) \) for any cha... | Proof. (1) Suppose \( B \in \Delta \) is a simplex such that \( \rho \left( B\right) = A \leq C \) . Choose an apartment \( {\sum }^{\prime \prime } \) containing both \( B \) and \( C \) . Then \( {\left. \rho \right| }_{{\sum }^{\prime \prime }} \) is an isomorphism, and it maps both \( A \) and \( B \) to \( A \) . ... | Yes |
Proposition 4.40. Every apartment \( \sum \) is a convex chamber subcomplex of \( \Delta \) . More precisely, given \( C, A \in \sum \) with \( C \) a chamber, every minimal gallery in \( \Delta \) between \( C \) and \( A \) is contained in \( \sum \) . | Proof. Let \( \Gamma : {C}_{0},\ldots ,{C}_{d} \) be a minimal gallery from \( C \) to \( A \) . If \( \Gamma \) is not contained in \( \sum \), then there is an index \( i \geq 1 \) with \( {C}_{i - 1} \in \sum \) and \( {C}_{i} \notin \sum \) . Let \( D \) be the chamber of \( \sum \) distinct from \( {C}_{i - 1} \) ... | Yes |
Proposition 4.41. Let \( \Gamma : {C}_{0},\ldots ,{C}_{d} \) be a gallery of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{d}}\right) \) . Then \( \Gamma \) is minimal if and only if \( \mathbf{s} \) is a reduced word. | Proof. If \( \Gamma \) is minimal, then it is contained in an apartment by Proposition \( {4.40};\mathrm{s} \) is then reduced by the connection between words in a Coxeter group and galleries in the associated Coxeter complex. Conversely, suppose \( \mathbf{s} \) is reduced. We may assume by induction that the subgalle... | Yes |
Corollary 4.42. Let \( C \) and \( D \) be chambers, and let \( \mathbf{s} \) be a reduced word. If there is a gallery from \( C \) to \( D \) of type \( \mathbf{s} \), then there is a unique such gallery. | Proof. Choose an apartment \( \sum \) containing \( C \) and \( D \) . Any gallery from \( C \) to \( D \) of type \( \mathbf{s} \) is minimal and hence is contained in \( \sum \) . The result now follows from the thinness of \( \sum \) . | No |
Proposition 4.44. Let \( \Delta \) be a connected bipartite graph in which every vertex is a face of at least two edges. Then \( \Delta \) is a building if and only if \( \Delta \) has diameter \( m \) and girth \( {2m} \) for some \( m \) with \( 2 \leq m \leq \infty \) . In this case \( \Delta \) has type \( {I}_{2}\... | Proof. Suppose that \( \Delta \) has diameter \( m \) and girth \( {2m} \) . Then one easily checks the following properties for any two vertices \( u, v \) :\n\n- If \( d\left( {u, v}\right) < m \), then there is a unique path from \( u \) to \( v \) of length \( \leq m \) with no backtracking. [\ | No |
Theorem 4.54. If \( \Delta \) is a building, then the union of any family of apartment systems is again an apartment system. Consequently, \( \Delta \) admits a largest system of apartments. | Proof. It is obvious that (B0) and (B1) hold for the union, so the only problem is to prove (B2). We will work with the variant \( \left( {\mathrm{B}{2}^{\prime \prime }}\right) \) . Suppose, then, that \( \sum \) and \( {\sum }^{\prime } \) are apartments in different apartment systems and that \( \sum \cap {\sum }^{\... | Yes |
Lemma 4.57. Let \( \mathbf{m} = {\left( {m}_{i}\right) }_{i \in I} \) and \( {\mathbf{m}}^{\prime } = {\left( {m}_{i}^{\prime }\right) }_{i \in I} \), where \( I \) is a finite index set and the elements \( {m}_{i},{m}_{i}^{\prime } \) are in some totally ordered set. Suppose that \( {\mathbf{m}}^{\prime } \) is a perm... | Sketch of proof. Let \( m \) be the smallest element that occurs in \( \mathbf{m} \), and let \( J \mathrel{\text{:=}} \left\{ {i \in I \mid {m}_{i} = m}\right\} \) . Then we have \( {m}_{i}^{\prime } \geq m \) for \( i \in J \) and \( {m}_{i}^{\prime } > m \) for \( i \notin J \) . But \( m \) must occur in \( {\mathb... | No |
Lemma 4.58. Let \( \sum \) be a subcomplex of \( \Delta \) that is isomorphic to \( {\sum }_{M} \), and let \( {M}^{\prime } \) be its Coxeter matrix as above. Then \( {M}^{\prime } = M \) . Consequently, there is a type-preserving isomorphism between \( \sum \) and \( {\sum }_{M} \) . | Proof. We know from Proposition 3.85 that there is a type-preserving isomorphism between \( \sum \) and \( {\sum }_{{M}^{\prime }} \) . So the second assertion follows from the first. To prove the first assertion, let \( \phi : \sum \rightarrow {\sum }_{M} \) be an isomorphism, and let \( f : S \rightarrow S \) be the ... | Yes |
Proposition 4.59. If \( \sum \) is a subcomplex of \( \Delta \) that is isomorphic to \( {\sum }_{M} \), then \( \sum \) is an apartment in the complete system of apartments. | Proof. It suffices to show that if \( \sum \) is adjoined to an apartment system \( \mathcal{A} \) , then axiom \( \left( {\mathrm{{B2}}}^{\prime \prime }\right) \) still holds. The proof is essentially the same as the proof of Theorem 4.54. The given complex \( \sum \) plays the role of the complex \( {\sum }^{\prime ... | No |
Proposition 4.63. Let \( \Delta \) be a building of type \( M \) . A chamber subcomplex \( {\Delta }^{\prime } \) of \( \Delta \) is a subbuilding if and only if \( {\Delta }^{\prime } \) is a building in its own right and its Coxeter matrix is \( M \) . | Proof. The \ | No |
Let \( \Delta = \Delta \left( {k}^{n}\right) \) be the building associated to an \( n \) -dimensional vector space over a field \( k \) (Section 4.3). If \( {k}^{\prime } \) is a subfield of \( k \), then there is an obvious embedding of \( {\Delta }^{\prime } \mathrel{\text{:=}} \Delta \left( {\left( {k}^{\prime }\rig... | One can see this directly from Definition 4.62 and the construction of apartments in Section 4.3. Alternatively, it follows at once from Proposition 4.63. | No |
Theorem 4.66. Let \( \Delta \) be a building and \( {\Delta }^{\prime } \) a chamber subcomplex of \( \Delta \) . Then \( {\Delta }^{\prime } \) is a subbuilding if and only if it is weak and is convex in \( \Delta \) . | Proof of the \ | No |
Lemma 4.69. Let \( C \) and \( {C}^{\prime } \) be opposite chambers in a spherical building, and let \( \sum \) be any apartment containing \( C \) and \( {C}^{\prime } \) . Then every chamber of \( \sum \) occurs in some minimal gallery from \( C \) to \( {C}^{\prime } \). | [This is the combinatorial analogue of the following geometric fact: Given two opposite points \( x,{x}^{\prime } \) of a sphere, the geodesics (great semicircles) from \( x \) to \( {x}^{\prime } \) cover the entire sphere.] | No |
Theorem 4.70. A spherical building \( \Delta \) admits a unique system of apartments. The apartments are precisely the convex hulls of pairs \( C,{C}^{\prime } \) of opposite chambers. | Proof. Let \( \mathcal{A} \) be an arbitrary system of apartments, and let \( {\mathcal{A}}^{\prime } \) be the set of convex hulls of pairs of opposite chambers. Every apartment \( \sum \in \mathcal{A} \) contains a pair of opposite chambers and is their convex hull, as we observed above; so \( \mathcal{A} \subseteq {... | Yes |
Theorem 4.73. If \( \Delta \) is a spherical building of rank \( n \), then \( \left| \Delta \right| \) has the homotopy type of a bouquet of \( \left( {n - 1}\right) \) -spheres, where there is one sphere for every apartment containing a fixed chamber \( C \) . | It remains to say something about the claim. Note first that every apartment \( \sum \) containing \( C \) admits a canonical type-preserving isomorphism to \( {\sum }_{M} \) , with \( C \) going to the fundamental chamber of \( {\sum }_{M} \) . Now \( \left| {\sum }_{M}\right| \) can be identified with the unit sphere... | No |
Proposition 4.81. There is a function \( \delta : \mathcal{C}\left( \Delta \right) \times \mathcal{C}\left( \Delta \right) \rightarrow W \) with the following properties:\n\n(1) Given a minimal gallery \( \Gamma : {C}_{0},\ldots ,{C}_{d} \) of type \( \mathbf{s}\left( \Gamma \right) = \left( {{s}_{1},\ldots ,{s}_{d}}\r... | Proof. Given chambers \( C, D \), choose an apartment \( \sum \) containing them and set \( \delta \left( {C, D}\right) = {\delta }_{\sum }\left( {C, D}\right) \) . This is independent of the choice of \( \sum \) by Proposition 4.6. Assertions (1) and (2) now follow from the convexity of apartments and the correspondin... | Yes |
Proposition 4.84. The Weyl distance function \( \delta : \mathcal{C}\left( \Delta \right) \times \mathcal{C}\left( \Delta \right) \rightarrow W \) has the following properties:\n\n(1) \( \delta \left( {C, D}\right) = 1 \) if and only if \( C = D \) .\n\n(2) \( \delta \left( {D, C}\right) = \delta {\left( C, D\right) }^... | Proof. (1) and (2) are immediate. To prove (3) and (4), let \( \delta \left( {C, D}\right) = w \) , let \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{d}}\right) \) be a reduced decomposition of \( w \), and choose a gallery \( C = {C}_{0},\ldots ,{C}_{d} = D \) of type s. If \( l\left( {sw}\right) = l\left( w\right) + 1... | Yes |
Theorem 4.86. Let \( \sum \) be a thin, convex chamber subcomplex of \( \Delta \) . Then \( \sum \) is an apartment in the complete apartment system. | Proof. By Proposition 4.59, it suffices to show that \( \sum \cong \sum \left( {W, S}\right) \) . Note that both \( \sum \) and \( \sum \left( {W, S}\right) \) are determined by their underlying chamber system, as in Section A.1.4. For \( \sum \left( {W, S}\right) \), we already observed this in Section 3.2. For \( \su... | No |
Proposition 4.88. Given two simplices \( A, B \in \Delta \), there is an element \( \delta \left( {A, B}\right) \in W \) such that\n\n\[ \delta \left( {A, B}\right) = \delta \left( {{C}_{0},{C}_{d}}\right) \]\n\nfor any minimal gallery \( {C}_{0},\ldots ,{C}_{d} \) from \( A \) to \( B \) . | Proof. Choose an apartment \( \sum \) containing \( A \) and \( B \), and set \( \delta \left( {A, B}\right) = \) \( {\delta }_{\sum }\left( {A, B}\right) \) . This is independent of the choice of \( \sum \) by Proposition 4.6. Equation (4.4) follows from Proposition 3.87, since the given gallery is a minimal gallery f... | Yes |
Proposition 4.95. Given a simplex \( A \) and a chamber \( C \), let \( {AC} \) be the product of \( A \) and \( C \) in any apartment containing \( A \) and \( C \) . Then \( {AC} \) is independent of the choice of apartment, and every minimal gallery from \( A \) to \( C \) starts with AC. Moreover, the gate property... | Proof. Choose a minimal gallery \( \Gamma : {C}_{0},\ldots ,{C}_{d} \) from \( A \) to \( C \) . If \( \sum \) is any apartment containing \( A \) and \( C \), then \( \Gamma \) is contained in \( \sum \) by Proposition 4.40; hence \( {C}_{0} \) is equal to the product \( {AC} \) defined in \( \sum \) (Section 3.6.4). ... | Yes |
Theorem 4.97. Let \( A \) be a simplex of cotype \( J \), let \( B \) be a simplex of cotype \( K \), and let \( w \) be the Weyl distance \( \delta \left( {A, B}\right) \) . Then \( {\mathcal{C}}_{A, B} \) is a residue of type \( {J}_{1} \mathrel{\text{:=}} J \cap {wK}{w}^{-1} \) . In other words, there is a simplex \... | Proof. Given \( {C}_{1},{C}_{2} \in {\mathcal{C}}_{A, B} \), let \( {D}_{i} = B{C}_{i} \) for \( i = 1,2 \) . Then \( {C}_{1} = A{D}_{1} \) by Proposition 4.96, so any apartment \( \sum \) containing \( {C}_{2} \) and \( {D}_{1} \) also contains \( {C}_{1} \) [because it contains \( A \) ]. Similarly, \( \sum \) contai... | Yes |
Proposition 4.101. For any two apartments \( \sum ,{\sum }^{\prime } \) there is a type-preserving isomorphism \( \phi : \sum \rightarrow {\sum }^{\prime } \) fixing \( \sum \cap {\sum }^{\prime } \) pointwise. | Proof. The proof is similar in spirit to the proof of \( \left( {\mathrm{{B2}}}^{\prime \prime }\right) \) in Section 4.1. Let \( M \) be a maximal simplex in \( \sum \cap {\sum }^{\prime } \) . Choose chambers \( C \in \sum \) and \( {C}^{\prime } \in {\sum }^{\prime } \) having \( M \) as a face. For each simplex \( ... | Yes |
Proposition 4.103. Let \( C \) be a chamber of a spherical building \( \Delta \) and let \( P \) be a panel in \( \Delta \) that is opposite some panel of \( C \). Among the chambers \( D > P \), there is a unique one that is not opposite \( C \), namely, the chamber \( D = {PC} \). | Proof. Note first that \( {PC} \) is not opposite \( C \), since in any apartment containing \( C \) and \( P \), the wall \( \operatorname{supp}P \) containing \( P \) does not separate \( C \) from \( {PC} \). Now consider any chamber \( D > P \) other than \( {PC} \). Any apartment \( \sum \) containing \( C \) and ... | Yes |
Proposition 4.104. Let \( \Delta \) be a thick spherical building. For any two chambers of \( \Delta \), there is a chamber that is opposite both of them. | Remark 4.105. This is a simple application of Proposition 4.103 if the two given chambers \( C, D \) are adjacent. To see this, let \( P \) be the common panel \( C \cap D \), and choose, by thickness, a chamber \( E > P \) different from \( C, D \) . Choose an apartment \( \sum \) containing \( C \) and \( E \), and l... | No |
Lemma 4.106. Let \( C \) and \( D \) be chambers in a spherical building. If \( C \) and \( D \) are not opposite, then there is a panel \( P < C \) such that \( C = {PD} \) . | Proof. This is actually a special case of Exercise 4.77 (see also part (b) of Exercise 1.59), but we will give an independent proof. Let \( {D}^{\prime } \) be the chamber opposite \( D \) in an apartment containing \( C \) and \( D \), and choose a minimal gallery from \( {D}^{\prime } \) to \( D \) passing through \(... | Yes |
Lemma 4.107. Let \( C, D \), and \( E \) be chambers in a thick spherical building. If \( E \) is not opposite \( D \), then there is a chamber \( {E}^{\prime } \) adjacent to \( E \) such that \( d\left( {{E}^{\prime }, D}\right) > d\left( {E, D}\right) \) and \( d\left( {{E}^{\prime }, C}\right) \geq d\left( {E, C}\r... | Proof. By Lemma 4.106, there is a panel \( P < E \) such that \( E = {PD} \) . By thickness, there is chamber \( {E}^{\prime } > P \) different from both \( {PC} \) and \( {PD} = E \) . Using the gate property, we find that\n\n\[ d\left( {{E}^{\prime }, D}\right) = d\left( {{PD}, D}\right) + 1 = d\left( {E, D}\right) +... | Yes |
Proposition 4.115. If \( {\Delta }^{\prime } \) is a subcomplex of \( \Delta \) that contains at least one chamber, then the following conditions are equivalent:\n\n(i) \( {\Delta }^{\prime } \) is a convex chamber subcomplex of \( \Delta \) .\n\n(ii) \( {\Delta }^{\prime } \) is closed under products, i.e., for any \(... | Proof. The arguments given in Section 3.6.6 for Coxeter complexes (see Lemma 3.130 and Theorem 3.131) extend to buildings, with no essential change. | No |
Suppose \( \alpha \) is a root of a spherical building \( \Delta \) . Let \( P \) be a panel in \( \partial \alpha \), let \( {P}^{\prime } \) be the panel in \( \partial \alpha \) opposite \( P \), and let \( {C}^{\prime } \) be the chamber of \( \alpha \) having \( {P}^{\prime } \) as a face. Then \( \Gamma \left( {P... | To see this, choose an apartment \( \sum \) containing \( \alpha \) . As we noted above, \( \Gamma \left( {P,{C}^{\prime }}\right) \) coincides with the convex hull of \( P \) and \( {C}^{\prime } \) in \( \sum \) . Our assertion now follows from Example 3.133(d). | No |
Lemma 4.118. Let \( \alpha \) be a root in a spherical building \( \Delta \), and let \( P \) be a panel in \( \partial \alpha \) . Then there is a canonical bijection from \( \mathcal{C}\left( {P,\alpha }\right) \) to \( \mathcal{A}\left( \alpha \right) \) . It associates to any chamber \( D \in \mathcal{C}\left( {P,\... | Proof. Let \( C \) be the chamber in \( \alpha \) having \( P \) as a face, let \( {P}^{\prime } \) be the panel opposite \( P \) in \( \alpha \), and let \( {C}^{\prime } \) be the chamber in \( \alpha \) having \( {P}^{\prime } \) as a face. By Proposition 4.103, \( C \) is the unique chamber in \( {\mathcal{C}}_{P} ... | Yes |
Proposition 4.121. Every convex subcomplex \( {\Delta }^{\prime } \) of \( \Delta \) is a chamber complex. | Proof. Let \( A \) and \( B \) be maximal simplices of \( {\Delta }^{\prime } \), choose an apartment \( \sum \) containing them, and let \( {\sum }^{\prime } \mathrel{\text{:=}} \sum \cap {\Delta }^{\prime } \) . Then \( {\sum }^{\prime } \) is a convex subcomplex of \( \sum \) ; hence it is a chamber complex by Propo... | Yes |
Theorem 4.127. If \( \Delta \) is a spherical building of rank \( n \), then \( \left| \Delta \right| \) has the homotopy type of a bouquet of \( \left( {n - 1}\right) \) -spheres, where there is one sphere for every apartment containing \( C \) . If \( \Delta \) is a nonspherical building, then \( \left| \Delta \right... | Our outline of the proof will be complete as far as the theory of buildings is concerned, but we will omit some homotopy-theoretic details. The idea is to start with \( C \) and then keep track of the homotopy type as one successively adjoins the chambers adjacent to \( C \), then the chambers at distance 2 from \( C \... | No |
Lemma 4.128. Let \( \Delta \) be an arbitrary building. Fix a chamber \( C \) and an integer \( d \geq 1 \), and let \( \mathcal{D} \) be a set of chambers with the following two properties:\n\n(1) \( d\left( {C, D}\right) \leq d \) for every \( D \in \mathcal{D} \).\n\n(2) \( \mathcal{D} \) contains every chamber \( D... | Proof of Lemma 4.128. The first claim is that\n\n\[ \bar{D} \cap {\Delta }^{\prime } = \{ B < D \mid d\left( {C, B}\right) < d\} .\n\nThe right side is trivially contained in the left side by hypothesis (2). To prove the opposite inclusion, suppose \( B \in \bar{D} \cap {\Delta }^{\prime } \). Then there is a chamber \... | Yes |
Lemma 5.3. Let \( \left( {\mathcal{C},\delta }\right) \) be a building of type \( \left( {W, S}\right) \). (1) If \( C, D \in \mathcal{C} \) satisfy \( \delta \left( {C, D}\right) = s \in S \), then also \( \delta \left( {D, C}\right) = s \). | Proof. (1) Method 1: Set \( w \mathrel{\text{:=}} \delta \left( {D, C}\right) \). By (WD1), \( w \neq 1 \) and \( 1 = \delta \left( {C, C}\right) \). By (WD2), we have \( 1 = \delta \left( {C, C}\right) \in \{ {sw}, w\} \). Since \( w \neq 1 \), we must have \( {sw} = 1 \) and hence \( \delta \left( {D, C}\right) = w =... | Yes |
Lemma 5.5. The chamber \( {C}^{\prime } \) in (WD3) is uniquely determined if \( l\left( {sw}\right) = \) \( l\left( w\right) - 1 \) . | Proof. Let \( \mathcal{P} \) be the \( s \) -panel containing \( C \), and choose a chamber \( {C}^{\prime } \) as in (WD3). Then \( {C}^{\prime } \in \mathcal{P} \), and we are trying to prove that it is the unique chamber in \( \mathcal{P} \) such that \( \delta \left( {{C}^{\prime }, D}\right) = {w}^{\prime } \mathr... | Yes |
If we define \( {\delta }_{W} : W \times W \rightarrow W \) by \( {\delta }_{W}\left( {{w}_{1},{w}_{2}}\right) = {w}_{1}^{-1}{w}_{2} \), then \( \left( {W,{\delta }_{W}}\right) \) is a thin building of type \( \left( {W, S}\right) \) (cf. Section 3.5). So here the chambers are elements of \( W \), and each \( x \in W \... | More generally, for any three chambers \( C, D, E \in W \) , we have\n\n\[ \n{\delta }_{W}\left( {C, E}\right) = {\delta }_{W}\left( {C, D}\right) {\delta }_{W}\left( {D, E}\right) .\n\] | Yes |
Lemma 5.16. Let \( C \) and \( D \) be chambers and let \( w \mathrel{\text{:=}} \delta \left( {C, D}\right) \) . (1) If \( \Gamma \) is a gallery of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{n}}\right) \) connecting \( C \) and \( D \), then there exists a subword \( \left( {{s}_{{i}_{1}},\ldots ,{s}_{{i}_{m}}... | Proof. In both parts of the lemma, we proceed by induction on \( n \) . (1) Let \( \Gamma : C = {C}_{0},{C}_{1},\ldots ,{C}_{n} = D \) be the given gallery. Let \( {w}^{\prime } = \) \( \delta \left( {{C}_{1}, D}\right) \), and apply (WD2) to the triangle to deduce that \( w \in \left\{ {{s}_{1}{w}^{\prime },{w}^{\prim... | Yes |
Corollary 5.17. For any two chambers \( C, D \in \mathcal{C} \), we have:\n\n(1) \( d\left( {C, D}\right) = l\left( {\delta \left( {C, D}\right) }\right) \) .\n\n(2) \( \delta \left( {D, C}\right) = \delta {\left( C, D\right) }^{-1} \) . | Proof. Set \( w \mathrel{\text{:=}} \delta \left( {C, D}\right) \) and choose a reduced decomposition \( w = {s}_{1}\cdots {s}_{n} \) of \( w \) . By Lemma 5.16(2), there exists a gallery \( \Gamma : C = {C}_{0},\ldots ,{C}_{n} = D \) of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{n}}\right) \) connecting \( C \)... | Yes |
Proposition 5.23. Let \( \left( {\mathcal{C},{\left( { \sim }_{s}\right) }_{s \in S}}\right) \) be a chamber system over \( S \) such that each panel contains at least two chambers. Then a map \( \delta : \mathcal{C} \times \mathcal{C} \rightarrow W \) satisfies (G) if and only if it satisfies (WD1)-(WD3). | Proof. It is shown in Lemma 5.16 that the WD axioms imply (G). So we assume that \( \delta \) satisfies (G) and deduce the WD axioms, which we may take in the form (WD1), (WD2a), (WD2b), and (WD3) (see Exercise 5.10).\n\n(WD1) By (G) applied to the empty word, two chambers \( C, D \) satisfy \( \delta \left( {C, D}\rig... | Yes |
Lemma 5.28. Given \( C, D, E \in \mathcal{C} \), set \( u \mathrel{\text{:=}} \delta \left( {C, D}\right) \) and \( v \mathrel{\text{:=}} \delta \left( {D, E}\right) \) . Then the following hold:\n\n(1) If \( u = {s}_{1}\cdots {s}_{m} \) with \( {s}_{i} \in S \) for all \( i \), then \( \delta \left( {C, E}\right) = {s... | Proof. (1) The argument is essentially the same as in Lemma 5.16(1): Choose a gallery \( \Gamma : C = {C}_{0},\ldots ,{C}_{m} = D \) of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{m}}\right) \) connecting \( C \) and \( D \) (this is possible by Lemma \( {5.16}\left( 2\right) \) ). Then \( \delta \left( {{C}_{m -... | Yes |
Lemma 5.29. Let \( \mathcal{R} \) be a \( J \)-residue and \( \mathcal{S} \) a \( K \)-residue with \( J, K \subseteq S \). Then \( \delta \left( {\mathcal{R},\mathcal{S}}\right) \) is a double coset of the form \( {W}_{J}w{W}_{K} \). In particular, \( \delta \left( {\mathcal{R},\mathcal{R}}\right) = {W}_{J} \). | Proof. The second assertion follows from the first since \( 1 \in \delta \left( {\mathcal{R},\mathcal{R}}\right) \). To prove the first assertion, choose \( C \in \mathcal{R} \) and \( D \in \mathcal{S} \) and set \( w = \delta \left( {C, D}\right) \). Given \( {C}^{\prime } \in \mathcal{R} \) and \( {D}^{\prime } \in ... | Yes |
Corollary 5.30. If \( J \subseteq S \) and \( \mathcal{R} \) is a \( J \) -residue, then \( \left( {\mathcal{R},{\left. \delta \right| }_{\mathcal{R} \times \mathcal{R}}}\right) \) is a building of type \( \left( {{W}_{J}, J}\right) \) . | Proof. We have \( \delta \left( {\mathcal{R},\mathcal{R}}\right) = {W}_{J} \) by the lemma, so the restricted function \( {\left. \delta \right| }_{\mathcal{R} \times \mathcal{R}} \) does indeed take values in \( {W}_{J} \) . The verification of (WD1)-(WD3) for \( \mathcal{R} \) is now straightforward and is left to th... | No |
Proposition 5.34. Let \( \mathcal{R} \) be a residue and \( D \) a chamber. Then there exists a unique \( {C}_{1} \in \mathcal{R} \) such that \( d\left( {{C}_{1}, D}\right) = d\left( {\mathcal{R}, D}\right) \) . This chamber \( {C}_{1} \) has the following properties:\n\n(1) \( \delta \left( {{C}_{1}, D}\right) = \min... | Proof. Choose \( {C}_{1} \in \mathcal{R} \) at minimal distance from \( D \) . Then \( {C}_{1} \) satisfies (i) by the discussion above, with \( \mathcal{S} = \{ D\} \) . Now let \( J \) be the type of \( \mathcal{R} \), and let \( {w}_{1} = \min \left( {\delta \left( {\mathcal{R}, D}\right) }\right) = \delta \left( {{... | Yes |
Lemma 5.36. Let \( \mathcal{R} \) and \( \mathcal{S} \) be residues of types \( J \) and \( K \), respectively, where \( J, K \subseteq S \) . Let \( {w}_{1} \mathrel{\text{:=}} \min \left( {\delta \left( {\mathcal{R},\mathcal{S}}\right) }\right) \) . (1) The projection \( \mathcal{P} \mathrel{\text{:=}} {\operatorname... | Proof. (1) Set \( {\mathcal{P}}^{\prime } = \left\{ {C \in \mathcal{R} \mid {w}_{1} \in \delta \left( {C,\mathcal{S}}\right) }\right\} \) . Suppose \( C \in \mathcal{R} \) and \( D \in \mathcal{S} \) satisfy \( \delta \left( {C, D}\right) = {w}_{1} \) or, equivalently, \( d\left( {C, D}\right) = d\left( {\mathcal{R},\m... | Yes |
Proposition 5.37. Let \( \mathcal{R} \) be a residue of type \( J \) and \( \mathcal{S} \) a residue of type \( K \) in \( \mathcal{C}\left( {J, K \subseteq S}\right) \) . Set \( {\mathcal{R}}_{1} = {\operatorname{proj}}_{\mathcal{R}}\mathcal{S} \) and \( {\mathcal{S}}_{1} = {\operatorname{proj}}_{\mathcal{S}}\mathcal{... | Proof. The first assertion of (1) is already contained in Lemma 5.36, and the second follows by reversing the roles of \( \mathcal{R} \) and \( \mathcal{S} \) . Now consider any \( C \in {\mathcal{R}}_{1} \) . Then, by the lemma, there exists \( D \in \mathcal{S} \) with \( d\left( {C, D}\right) = d\left( {\mathcal{R},... | Yes |
Lemma 5.45. Let \( \mathcal{M} \) be a convex subset of \( \mathcal{C} \), and let \( \mathcal{R} \) be a residue in \( \mathcal{C} \) that meets \( \mathcal{M} \) . Then for any chamber \( D \in \mathcal{M} \), we have \( {\operatorname{proj}}_{\mathcal{R}}D \in \mathcal{M} \) . | Proof. Choose \( C \in \mathcal{R} \cap \mathcal{M} \) and set \( {C}^{\prime } = {\operatorname{proj}}_{\mathcal{R}}D \) . By the gate property (Proposition 5.34), \( d\left( {C, D}\right) = d\left( {C,{C}^{\prime }}\right) + d\left( {{C}^{\prime }, D}\right) \) . This says precisely that there is a minimal gallery fr... | Yes |
Proposition 5.46. The following conditions on a nonempty subset \( \mathcal{M} \subseteq \mathcal{C} \) are equivalent.\n\n(i) \( \mathcal{M} \) is convex.\n\n(ii) \( {\operatorname{proj}}_{\mathcal{P}}D \in \mathcal{M} \) for every chamber \( D \in \mathcal{M} \) and every panel \( \mathcal{P} \) in \( \mathcal{C} \) ... | Proof. (i) \( \Rightarrow \) (ii) by Lemma 5.45. For the converse, suppose (ii) holds. Given a minimal gallery \( \Gamma : {C}_{0},\ldots ,{C}_{n} \) with \( {C}_{0},{C}_{n} \in \mathcal{M} \), we have to show that all \( {C}_{i} \) are in \( \mathcal{M} \) . Since \( \Gamma \) is minimal, \( d\left( {{C}_{i},{C}_{n}}\... | No |
Proposition 5.47. A nonempty subset \( \mathcal{M} \subseteq \mathcal{C} \) is convex if and only if \( \left( {\mathcal{M},{\delta }_{\mathcal{M}}}\right) \) has the following property: Given \( C, D \in \mathcal{M} \) and a reduced decomposition \( \mathbf{s} \) of \( \delta \left( {C, D}\right) \), there is a galler... | Proof. Suppose \( \mathcal{M} \) is convex. Given \( C, D \) in \( \mathcal{M} \) and a reduced decomposition \( \mathbf{s} \) of \( w \mathrel{\text{:=}} \delta \left( {C, D}\right) \), there is a minimal gallery from \( C \) to \( D \) in \( \mathcal{C} \) of type \( \mathbf{s} \) by Lemma 5.16. This gallery is conta... | Yes |
Proposition 5.52. A nonempty subset \( \mathcal{M} \subseteq \mathcal{C} \) is a subbuilding if and only if it is weak and convex. | Proof. Method 1: If \( \mathcal{M} \) is a subbuilding, then it is weak by (WD3) and it is convex by Proposition 5.47 (and Lemma 5.16). Conversely, if \( \mathcal{M} \) is weak and convex, we must show that it satisfies (WD3). Given \( C, D \in \mathcal{M} \) and \( s \in S \) , set \( w \mathrel{\text{:=}} \delta \lef... | Yes |
Lemma 5.55. If \( \mathcal{A} \) is a thin building, then for all \( C, D, E \in \mathcal{A} \), we have \( \delta \left( {C, E}\right) = \delta \left( {C, D}\right) \delta \left( {D, E}\right) . \) | Proof. Consider first the case \( \delta \left( {C, D}\right) = s \in S \), as in axiom (WD2). Since \( \mathcal{A} \) is thin, \( C \) is the only chamber in \( \mathcal{A} \) that is \( s \) -adjacent to \( D \) . Now (WD3) implies that some chamber \( {D}^{\prime } \in \mathcal{A} \) must satisfy \( \delta \left( {{... | Yes |
Proposition 5.58. A set of chambers is convex if and only if it is 2-convex and gallery connected. | Proof. The \ | No |
Lemma 5.61. Let \( \left( {\mathcal{C},\delta }\right) \) be a building of type \( \left( {W, S}\right) \), let \( \left( {{\mathcal{C}}^{\prime },{\delta }^{\prime }}\right) \) be a building of type \( \left( {{W}^{\prime },{S}^{\prime }}\right) \), and let \( \sigma : \left( {W, S}\right) \rightarrow \left( {{W}^{\pr... | Proof. The \ | No |
Lemma 5.62. Let \( \mathcal{C} \) be a building of type \( \left( {W, S}\right) ,{\mathcal{C}}^{\prime } \) a building of type \( \left( {{W}^{\prime },{S}^{\prime }}\right) \), and \( \sigma : \left( {W, S}\right) \rightarrow \left( {{W}^{\prime },{S}^{\prime }}\right) \) an isomorphism of Coxeter systems.\n\n(1) If \... | Proof. (1) This is an easy consequence of the convexity criterion given in Proposition 5.47. In detail, suppose we are given two chambers \( C, D \in \mathcal{M} \) and a reduced decomposition \( {\mathbf{s}}^{\prime } \) of \( {w}^{\prime } \mathrel{\text{:=}} \delta \left( {\phi \left( C\right) ,\phi \left( D\right) ... | Yes |
Proposition 5.65. If \( \left( {\mathcal{C},\delta }\right) \) is a thin building of type \( \left( {W, S}\right) \), then \( \mathcal{C} \) is isometric to \( W \), where the latter is viewed as the set of chambers of the standard thin building \( \left( {W,{\delta }_{W}}\right) \) . | Proof. Fix a chamber \( {C}_{0} \in \mathcal{C} \) and define \( \psi : \mathcal{A} \rightarrow W \) by \( \psi \left( C\right) \mathrel{\text{:=}} \delta \left( {{C}_{0}, C}\right) \) for \( C \in \mathcal{C} \) . Applying Corollary 5.17(2) and Lemma 5.55, we obtain, for all \( C, D \in \mathcal{C} \) ,\n\n\[{\delta }... | Yes |
Corollary 5.66. Let \( \left( {\mathcal{C},\delta }\right) \) be a thin building of type \( \left( {W, S}\right) \), and fix a chamber \( {C}_{0} \in \mathcal{C} \) . Then for any \( w \in W \), there is a unique chamber \( C \in \mathcal{C} \) such that \( \delta \left( {{C}_{0}, C}\right) = w. \) | The corollary is trivially true in the standard thin building (most obviously with \( {C}_{0} = 1 \), since \( {\delta }_{W}\left( {1, w}\right) = w \) ), and this is what motivated the definition of \( \psi \) in the proof of the proposition. | No |
Corollary 5.67. Let \( \left( {\mathcal{C},\delta }\right) \) be a building of type \( \left( {W, S}\right) \) and let \( \left( {W,{\delta }_{W}}\right) \) be the standard thin building. Then a subset \( \mathcal{A} \subseteq \mathcal{C} \) is an apartment of \( \mathcal{C} \) if and only if it is isometric to \( W \)... | Proof. If \( \mathcal{A} \) is an apartment, then it is a thin building and hence is isometric to \( W \) by Proposition 5.65. Conversely, if there exists an isometry \( \phi : W \rightarrow \mathcal{C} \) with \( \phi \left( W\right) = \mathcal{A} \), then \( \mathcal{A} \) is a subbuilding of \( \mathcal{C} \) by Lem... | Yes |
Corollary 5.68. Any two apartments \( {\mathcal{A}}_{1},{\mathcal{A}}_{2} \) of \( \mathcal{C} \) are isometric. Furthermore, we can always find a surjective isometry \( \phi : {\mathcal{A}}_{1} \rightarrow {\mathcal{A}}_{2} \) with \( \phi \left( C\right) = C \) for all \( C \in {\mathcal{A}}_{1} \cap {\mathcal{A}}_{2... | Proof. By Corollary 5.67, there exist surjective isometries \( {\phi }_{1} : W \rightarrow {\mathcal{A}}_{1} \) and \( {\phi }_{2} : W \rightarrow {\mathcal{A}}_{2} \), so \( \phi \mathrel{\text{:=}} {\phi }_{2} \circ {\phi }_{1}^{-1} : {\mathcal{A}}_{1} \rightarrow {\mathcal{A}}_{2} \) is a surjective isometry. If \( ... | Yes |
Lemma 5.71. Let \( \mathcal{C} \) be a building of type \( \left( {W, S}\right) \), let \( C,{D}_{1},{D}_{2} \) be chambers in \( \mathcal{C} \), and let \( s \in S \) satisfy \( l\left( {{s\delta }\left( {C,{D}_{i}}\right) }\right) < l\left( {\delta \left( {C,{D}_{i}}\right) }\right) \) for \( i = 1,2 \) . Suppose tha... | Proof of the lemma. Set \( {w}_{i} = \delta \left( {C,{D}_{i}}\right) ,{v}_{i} = s{w}_{i} \), and \( {C}_{i} = {\operatorname{proj}}_{\mathcal{P}}{D}_{i} \) for \( i = 1,2 \) . Since \( l\left( {v}_{i}\right) < l\left( {w}_{i}\right) \), we have \( {v}_{i} = \delta \left( {{C}_{i},{D}_{i}}\right) ,{C}_{i} \neq C \), an... | Yes |
Corollary 5.74. For any two chambers \( C, D \in \mathcal{C} \), there exists an apartment \( \mathcal{A} \) of \( \mathcal{C} \) with \( C, D \in \mathcal{A} \) . | Proof. This follows from the second assertion of the theorem, since \( \{ C, D\} \) is isometric to \( \{ 1, w\} \), where \( w = \delta \left( {C, D}\right) \) . | No |
Proposition 5.81. Let \( \alpha \) be a root of \( \mathcal{C} \) . Then we have:\n\n(1) \( \alpha \) is a convex subset of \( \mathcal{C} \) .\n\n(2) \( \alpha \) is contained in an apartment of \( \mathcal{C} \) .\n\n(3) If \( \mathcal{C} \) is the standard thin building of type \( \left( {W, S}\right) \), then \( \a... | Proof. By definition, there is an isometry \( \phi : {\alpha }_{s} \rightarrow \mathcal{C} \) for some \( s \in S \), with image \( \alpha \) . Since \( {\alpha }_{s} \) is a convex subset of \( W \) by Lemma 3.44, it follows from Lemma 5.62(1) that \( \alpha \) is a convex subset of \( \mathcal{C} \) . This proves (1)... | Yes |
Lemma 5.85. Let \( \\mathcal{R} \) and \( \\mathcal{S} \) be two residues in \( \\mathcal{C} \) of respective types \( J \) and \( K \). (1) There is an inclusion-preserving bijection from the set of residues of \( \\mathcal{C} \) containing \( \\mathcal{R} \) to the set of subsets of \( S \) containing \( J \). It ass... | Proof. (1) If we fix \( C \\in \\mathcal{R} \), then \( {J}^{\\prime } \\mapsto {R}_{{J}^{\\prime }}\\left( C\\right) \) defines an inclusion-preserving map from \( \\left\\{ {{J}^{\\prime } \\mid J \\subseteq {J}^{\\prime } \\subseteq S}\\right\\} \) to the set of residues of \( \\mathcal{C} \) containing \( \\mathcal... | Yes |
Lemma 5.86. If \( {\mathcal{C}}^{\prime } \) is a subbuilding of \( \mathcal{C} \), then each residue \( {\mathcal{R}}^{\prime } \) of \( {\mathcal{C}}^{\prime } \) is contained in a smallest residue \( \mathcal{R} \) of \( \mathcal{C} \) . The map \( {\mathcal{R}}^{\prime } \mapsto \mathcal{R} \) is an inclusion-prese... | Proof. Denote the type of \( {\mathcal{R}}^{\prime } \) by \( J \) and choose \( C \in {\mathcal{R}}^{\prime } \) . Then each residue of \( \mathcal{C} \) containing \( {\mathcal{R}}^{\prime } \) must also contain \( \mathcal{R} \mathrel{\text{:=}} {R}_{J}\left( C\right) \), which is therefore the smallest residue of \... | Yes |
Lemma 5.88. \( \Delta \left( \mathcal{C}\right) \) is a colorable chamber complex of rank equal to \( \left| S\right| \) . Its set of chambers \( \mathcal{C}\left( {\Delta \left( \mathcal{C}\right) }\right) \) is equal to \( \mathcal{C} \), and the function \( \tau \) defined in (5.8) is a type function. If \( {\mathca... | Proof. We first check that the poset \( \Delta \mathrel{\text{:=}} \Delta \left( \mathcal{C}\right) \) has the properties (a) and (b) that characterize simplicial complexes (Definition A.1). By definition of \( \Delta \), two simplices \( A = {F}_{\mathcal{R}} \) and \( B = {F}_{\mathcal{S}} \) have a greatest lower bo... | Yes |
If \( \left( {W,{\delta }_{W}}\right) \) is the standard thin building of type \( \left( {W, S}\right) \), then \( \Delta \left( W\right) \) is equal to the standard Coxeter complex \( \sum \left( {W, S}\right) \) introduced in Chapter 3. | Indeed, the \( J \) -residues in \( W \) are the standard cosets \( w{W}_{J} \) ; hence, up to notation, \( \Delta \left( W\right) \) is the set of standard cosets, ordered by reverse inclusion, and that was precisely our definition of \( \sum \left( {W, S}\right) \) . | Yes |
Theorem 5.91. If \( \left( {\mathcal{C},\delta }\right) \) is a building of type \( \left( {W, S}\right) \), then \( \Delta \left( \mathcal{C}\right) \) is a simplicial building of type \( \left( {W, S}\right) \) in the sense of Definition 4.37. | Proof. We already verified in Lemma 5.88 that \( \Delta \left( \mathcal{C}\right) \) is a chamber complex, and we have given it a type function with values in \( S \) . Set\n\n\[ \Omega \mathrel{\text{:=}} \{ \Delta \left( \mathcal{A}\right) \mid \mathcal{A}\text{ is an apartment of }\mathcal{C}\} . \]\n\nRecall that w... | Yes |
Proposition 5.94. Let \( \Delta \) be a simplicial building, and let \( {\Delta }^{\prime } \) be a chamber subcomplex that is weak and convex. Then \( {\Delta }^{\prime } \) is a subbuilding of \( \Delta \) in the sense of Definition 4.62. | Proof. We may assume that \( \Delta \) comes equipped with a type function, so that it is a building of type \( \left( {W, S}\right) \) for some Coxeter system \( \left( {W, S}\right) \) . Then \( {\mathcal{C}}^{\prime } \mathrel{\text{:=}} \mathcal{C}\left( {\Delta }^{\prime }\right) \) is a weak and convex subset of ... | Yes |
Proposition 5.95. Let \( {\Delta }^{\prime } \) and \( \Delta \) be simplicial buildings of respective types \( \left( {{W}^{\prime },{S}^{\prime }}\right) \) and \( \left( {W, S}\right) \) . Let \( \psi : {\Delta }^{\prime } \rightarrow \Delta \) be an injective chamber map. If \( \psi \) is bijective, or if \( \sum \... | Proof. Recall that \( \Delta \) comes equipped with a type function having values in \( S \) , such that the diameter of \( {\operatorname{lk}}_{\Delta }A \) is equal to the order \( m\left( {s, t}\right) \) of \( {st} \) for all simplices \( A \) of cotype \( \{ s, t\} \subseteq S \), and similarly for \( {\Delta }^{\... | Yes |
Lemma 5.108. Let \( \mathcal{R} \) be a \( J \) -residue and \( D \) a chamber of \( \mathcal{C} \) . (1) \( \max \{ d\left( {C, D}\right) \mid C \in \mathcal{R}\} = {d}_{0}\left( J\right) + d\left( {\mathcal{R}, D}\right) \) . (2) \( d\left( {\mathcal{R}, D}\right) \leq {d}_{0} - {d}_{0}\left( J\right) \), with equali... | Proof. Let \( {C}^{\prime } \mathrel{\text{:=}} {\operatorname{proj}}_{\mathcal{R}}D \) . Then by the gate property (Proposition 5.34) we have \[ d\left( {C, D}\right) = d\left( {C,{C}^{\prime }}\right) + d\left( {{C}^{\prime }, D}\right) \] for all \( C \in \mathcal{R} \) . This is maximal when \( d\left( {C,{C}^{\pri... | Yes |
Proposition 5.109. Let \( \mathcal{R} \) be a \( J \) -residue, and let \( \mathcal{S} \) be a residue with \( \operatorname{rank}\mathcal{S} \geq \operatorname{rank}\mathcal{R} \) . Then the following conditions are equivalent.\n\n(i) \( \mathcal{R} \) op \( \mathcal{S} \) .\n\n(ii) \( d\left( {\mathcal{R},\mathcal{S}... | Proof. It is immediate from the definitions that the maximum on the right side of (iii) is equal to\n\n\[ \max \{ d\left( {\mathcal{R}, D}\right) \mid D \in \mathcal{C}\} \]\n\nBy Lemma 5.108, this maximum is equal to \( {d}_{0} - {d}_{0}\left( J\right) \) . So (ii) and (iii) are equivalent. The last assertion of Lemma... | Yes |
Lemma 5.111. Let \( \\left( {W,\\delta }\\right) \) be the standard thin building of type \( \\left( {W, S}\\right) \) (where \( W \) is finite). Then the following hold:\n\n(1) Any \( w \\in W \) is opposite precisely one \( {w}^{\\prime } \\in W \), namely \( {w}^{\\prime } = w{w}_{0} \). | Proof. (1) Given \( w,{w}^{\\prime } \\in W \), we have \( w \) op \( {w}^{\\prime } \\Leftrightarrow \\delta \\left( {w,{w}^{\\prime }}\\right) = {w}_{0} \\Leftrightarrow \) \( {w}^{-1}{w}^{\\prime } = {w}_{0} \\Leftrightarrow {w}^{\\prime } = w{w}_{0}. | Yes |
Lemma 5.113. Let \( \mathcal{A} \) be a thin spherical building. Then the opposition involution maps every root \( \alpha \) of \( \mathcal{A} \) to the opposite root \( - \alpha \) . | Proof. This can be proved in many ways. For example, one can identify \( \mathcal{A} \) with the set of chambers of a finite reflection group and use the fact that the opposition involution is given by multiplication by -1 . Alternatively, one can use the fact that opposite chambers in a spherical Coxeter complex are s... | No |
Proposition 5.114. If \( \mathcal{R} \) and \( \mathcal{S} \) are opposite residues of \( \mathcal{C} \), then \( {\operatorname{proj}}_{\mathcal{R}}\mathcal{S} = \mathcal{R} \) and \( {\operatorname{proj}}_{\mathcal{S}}\mathcal{R} = \mathcal{S} \) . | Proof. Let \( J \) be the type of \( \mathcal{R} \) and let \( K \) be the type of \( \mathcal{S} \) . By Lemma 5.107, \( \delta \left( {\mathcal{R},\mathcal{S}}\right) = {W}_{J}{w}_{0} = {w}_{0}{W}_{K} \) . This implies that \( \delta \left( {C,\mathcal{S}}\right) = {w}_{0}{W}_{K} = \delta \left( {\mathcal{R},\mathcal... | Yes |
Corollary 5.116. Let \( \mathcal{R} \) and \( \mathcal{S} \) be opposite residues in \( \mathcal{C} \), and let \( J \) be the type of \( \mathcal{R} \) . Then the projection map \( {\operatorname{proj}}_{\mathcal{S}} \) induces a surjective \( \sigma \) -isometry from \( \mathcal{R} \) onto \( \mathcal{S} \), where \(... | Proof. Since \( {\operatorname{proj}}_{\mathcal{R}}\mathcal{S} = \mathcal{R} \) and \( {\operatorname{proj}}_{\mathcal{S}}\mathcal{R} = \mathcal{S} \), Proposition 5.37 implies that the projection map from \( \mathcal{R} \) onto \( \mathcal{S} \) is a \( \sigma \) -isometry, with the projection from \( \mathcal{S} \) o... | Yes |
Corollary 5.117. If \( \mathcal{P} \) and \( \mathcal{Q} \) are opposite panels, then the relation of nonopposition induces a bijection between \( \mathcal{P} \) and \( \mathcal{Q} \) . | Proof. Given \( C \in \mathcal{P} \) and \( D \in \mathcal{Q} \), we have \( C \) op \( D \) if and only if \( d\left( {C, D}\right) = {d}_{0} \) . Otherwise \( d\left( {C, D}\right) = {d}_{0} - 1 \), and \( C \) and \( D \) correspond under the bijection of Corollary 5.116. In other words, \( C \) and \( D \) correspo... | Yes |
Corollary 5.118. If the spherical building \( \mathcal{C} \) is thick, then residues of \( \mathcal{C} \) of the same type are isometric. | Proof. Let \( \mathcal{R} \) and \( \mathcal{T} \) be two \( J \) -residues of \( \mathcal{C} \) . Choose arbitrary chambers \( C \in \mathcal{R} \) and \( D \in \mathcal{T} \) . By Proposition 4.104, there exists a chamber \( E \in \mathcal{C} \) that is opposite both \( C \) and \( D \) . [This is where we use thickn... | Yes |
Corollary 5.120. Let \( A \) and \( B \) be simplices of a simplicial spherical building \( \Delta \) of type \( \left( {W, S}\right) \). (1) If \( A \) and \( B \) are opposite, then \( {\operatorname{lk}}_{\Delta }A \) and \( {\operatorname{lk}}_{\Delta }B \) are isomorphic simplicial complexes. (2) If \( \Delta \) i... | Proof. Let \( \left( {\mathcal{C},\delta }\right) \) be the W-metric building of type \( \left( {W, S}\right) \) associated with \( \Delta \) (so \( \mathcal{C} = \mathcal{C}\left( \Delta \right) \). (1) By assumption, the residues \( {\mathcal{C}}_{ \geq A} \) and \( {\mathcal{C}}_{ \geq B} \) are opposite. So by Coro... | Yes |
Lemma 5.121. If \( w \) and \( {w}^{\prime } \) are opposite chambers of the standard thin building \( \left( {W,{\delta }_{W}}\right) \), then \( W \) is the convex hull of \( \left\{ {w,{w}^{\prime }}\right\} \) . | Proof. This is essentially a reformulation of the equation\n\n\[ \nl\left( {w}_{0}\right) = l\left( x\right) + l\left( {{x}^{-1}{w}_{0}}\right) \]\n\n(5.9)\n\nfor all \( x \in W \), which follows from Corollary 2.19. Here are the details. Given \( v \in W \), we want to show that there is a minimal gallery from \( w \)... | Yes |
Theorem 5.122. Let \( \left( {\mathcal{C},\delta }\right) \) be a spherical building.\n\n(1) If \( C \) and \( {C}^{\prime } \) are opposite chambers of \( \mathcal{C} \), then there is precisely one apartment \( \mathcal{A} \) of \( \mathcal{C} \) containing them both, and \( \mathcal{A} \) is the convex hull of \( \l... | Proof. Recall that apartments are convex subsets of \( \mathcal{C} \) by Corollary 5.54 and are isometric images of \( \left( {W,{\delta }_{W}}\right) \) by Corollary 5.67, where \( \left( {\mathcal{C},\delta }\right) \) is of type \( \left( {W, S}\right) \) .\n\n(1) There exists an apartment \( \mathcal{A} \) containi... | Yes |
Lemma 5.125. The dual \( \left( {\mathcal{C},{\delta }_{ - }}\right) \) of \( \left( {\mathcal{C},\delta }\right) \) is also a building of type \( \left( {W, S}\right) \), and it is \( {\sigma }_{0} \) -isometric to \( \left( {\mathcal{C},\delta }\right) \). The associated simplicial buildings \( \Delta \left( \mathcal... | Proof. Observe that \( {\delta }_{ - } = {\delta }^{{\sigma }_{0}} \) with the notation introduced in Exercise 5.64. So it follows from this exercise (which was a straightforward verification) that \( \left( {\mathcal{C},\delta }\right) \) is a building of type \( \left( {W, S}\right) \). By the very definition of the ... | Yes |
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