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Corollary 1.70. Let \( w = {s}_{1}{s}_{2}\cdots {s}_{m} \) with \( {s}_{i} \in S \). If there exists a shorter expression for \( w \) as a product of elements of \( S \), then there are indices \( i < j \) such that\n\n\[ w = {s}_{1}\cdots {\widehat{s}}_{i}\cdots {\widehat{s}}_{j}\cdots {s}_{m} \] | Proof. The hypothesis implies that \( d\left( {C,{wC}}\right) < m \), so the gallery corresponding to the given expression for \( w \) is not minimal. The conclusion now follows from the proof of (c) above. | No |
Corollary 1.72. \( \mathcal{H} \) necessarily consists of all hyperplanes \( H \) in \( V \) such that \( {s}_{H} \in W \) . | Proof. Suppose \( {s}_{H} \in W \) but \( H \notin \mathcal{H} \) . Then \( H \nsubseteq \mathop{\bigcup }\limits_{{{H}^{\prime } \in \mathcal{H}}}{H}^{\prime } \), so \( H \) must meet a chamber \( D \) . Since the element \( w \mathrel{\text{:=}} {s}_{H} \) of \( W \) fixes \( H \), it follows that \( {wD} \) meets \... | Yes |
Corollary 1.75. The chamber graph of \( \sum \left( {W, V}\right) \) is isomorphic, as a graph with \( W \) -action, to the Cayley graph of \( \left( {W, S}\right) \) . For any \( w \in W \), there is a \( 1 - 1 \) correspondence between galleries from \( C \) to \( {wC} \) and decompositions of \( w \) as an \( S \) -... | Proof. The bijection \( {wC} \leftrightarrow w \) sets up the isomorphism on the level of vertices. The remaining details should be clear at this point and are left to the reader. | No |
Proposition 1.77. Let \( \left( {W, V}\right) \) be a finite reflection group.\n\n(1) \( W \) has a unique element \( {w}_{0} \) of maximal length. Its length is given by \( l\left( {w}_{0}\right) = \left| \mathcal{H}\right| \)\n\n(2) The element \( {w}_{0} \) is characterized by the property that \( {w}_{0}C = - C \),... | Proof. By Theorem 1.69, there is a unique \( {w}_{0} \in W \) such that \( {w}_{0}C = - C \) . Parts (1) and (2) now follow at once from Proposition 1.57 and equation (1.15).\n\n(3) We have\n\n\[ l\left( {w{w}_{0}}\right) = d\left( {C, w{w}_{0}C}\right) \]\n\n\[ = d\left( {C, w\left( {-C}\right) }\right) \]\n\n\[ = d\l... | Yes |
Assume that \( \left( {W, V}\right) \) has rank 2 but is not necessarily essential. In other words, if we write \( V = {V}_{0} \oplus {V}_{1} \) as in Section 1.1, then \( \dim {V}_{1} = 2 \) . | By the previous example applied to \( \left( {W,{V}_{1}}\right) \), we have \( W \cong {D}_{2m} \) for some \( m \geq 2 \) . Moreover, if \( {C}_{1} \) is a chamber in \( {V}_{1} \) with walls \( {L}_{i} \) and normals \( {e}_{i} \) as above, then \( {V}_{0} \times {C}_{1} \) is a chamber in \( V \) with walls \( {V}_{... | Yes |
Example 1.81. (Type \( {\mathrm{A}}_{n - 1} \) ) Let \( W \) be the symmetric group on \( n \) letters acting on \( {\mathbb{R}}^{n} \) as in Example 1.10. Thus a permutation \( \pi \) acts by \( \pi \left( {v}_{i}\right) = {v}_{\pi \left( i\right) } \) for \( 1 \leq i \leq n \), where \( {v}_{1},\ldots ,{v}_{n} \) is ... | Indeed, we have \( \pi \left( {\mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}{v}_{i}}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}\pi \left( {v}_{i}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{x}_{i}{v}_{\pi \left( i\right) } \), whence (1.18). | Yes |
Example 1.82. (Type \( {\mathrm{C}}_{n} \) ) Let \( W \) be the signed permutation group acting on \( {\mathbb{R}}^{n} \) as in Example 1.11. Then \( \mathcal{H} \) consists of the hyperplanes \( {x}_{i} - {x}_{j} = 0 \) \( \left( {i \neq j}\right) ,{x}_{i} + {x}_{j} = 0\left( {i \neq j}\right) \), and \( {x}_{i} = 0 \... | \[ {\epsilon }_{1}{x}_{\pi \left( 1\right) } > {\epsilon }_{2}{x}_{\pi \left( 2\right) } > \cdots > {\epsilon }_{n}{x}_{\pi \left( n\right) } > 0 \] with \( {\epsilon }_{i} \in \{ \pm 1\} \) and \( \pi \in {S}_{n} \) . As fundamental chamber we can take \[ {x}_{1} > {x}_{2} > \cdots > {x}_{n} > 0. \] | Yes |
Example 1.83. (Type \( {\mathrm{D}}_{n} \) ) Let \( W \) be the subgroup of the signed permutation group consisting of elements that change an even number of signs (Example 1.13). Then \( \mathcal{H} \) consists of the hyperplanes \( {x}_{i} - {x}_{j} = 0 \) and \( {x}_{i} + {x}_{j} = 0\left( {i \neq j}\right) \) . To ... | \[ {\epsilon }_{1}{x}_{\pi \left( 1\right) } > {\epsilon }_{2}{x}_{\pi \left( 2\right) } > \cdots > {\epsilon }_{n - 1}{x}_{\pi \left( {n - 1}\right) } > \left| {x}_{\pi \left( n\right) }\right| \] (1.21) with \( {\epsilon }_{i} \in \{ \pm 1\} \) and \( \pi \in {S}_{n} \) . Note that the last inequality is equivalent t... | Yes |
Lemma 1.89. Given \( s \neq t \) in \( S \), let \( {W}^{\prime } \) be the group generated by \( s \) and \( t \) . Then \( {W}^{\prime } \) is a rank-2 reflection group, and \( C \) is contained in a \( {W}^{\prime } \) -chamber \( {C}^{\prime } \) having \( {H}_{s} \) and \( {H}_{t} \) as its walls. | Proof. We have \( {V}^{{W}^{\prime }} = {H}_{s} \cap {H}_{t} = {\left( \mathbb{R}{e}_{s} \oplus \mathbb{R}{e}_{t}\right) }^{ \bot } \), so \( \left( {{W}^{\prime }, V}\right) \) has rank 2 . Let \( {\mathcal{H}}^{\prime } \subseteq \mathcal{H} \) be the set of hyperplanes of the form \( {w}^{\prime }{H}_{s} \) or \( {w... | Yes |
Corollary 1.90. Assume that \( \left( {W, V}\right) \) is essential. Then \( \left( {W, V}\right) \) is reducible if and only if there is a partition of \( S \) into (nonempty) subsets \( {S}^{\prime },{S}^{\prime \prime } \) such that \( m\left( {s, t}\right) = 2 \) for all \( s \in {S}^{\prime } \) and \( t \in {S}^{... | Proof. Suppose there is such a partition. Let \( {W}^{\prime } \) and \( {W}^{\prime \prime } \) be the subgroups \( \left\langle {S}^{\prime }\right\rangle \) and \( \left\langle {S}^{\prime \prime }\right\rangle \), and let \( {V}^{\prime } \) (respectively \( {V}^{\prime \prime } \) ) be the subspace of \( V \) span... | No |
Corollary 1.94. Assume that \( \left( {W, V}\right) \) is essential. Then \( \left( {W, V}\right) \) is completely determined, up to isomorphism, by the matrix \( M \mathrel{\text{:=}} {\left( m\left( s, t\right) \right) }_{s, t \in S} \) . | Proof. Given \( M \), we can recover \( \left( {W, V}\right) \) as follows: \( V \) can be identified with \( {\mathbb{R}}^{S} \), the vector space of \ | No |
Theorem 1.104. Let \( \left( {W, V}\right) \) be a finite reflection group, \( C \) a chamber, and \( S \) the set of reflections with respect to the walls of \( C \) . Then \( \bar{C} \) is a strict fundamental domain for the action of \( W \) on \( V \) . Moreover, the stabilizer \( {W}_{x} \) of a point \( x \in \ba... | Proof. Since \( W \) is transitive on the chambers, it is clear that every point of \( V \) is \( W \) -equivalent to a point of \( \bar{C} \) . Everything else in the theorem will follow if we prove the following claim: For \( x, y \in \bar{C} \) and \( w \in W \), if \( {wx} = y \) then \( x = y \) and \( w \in \left... | Yes |
Corollary 1.105. For any cell \( A \), the stabilizer \( {W}_{A} \) of \( A \) (as a set) fixes \( A \) pointwise. | Proof. We may assume that \( A \) is a face of the fundamental chamber and hence that \( A \subseteq \bar{C} \) . Then no two distinct points of \( A \) are \( W \) -equivalent, and the result follows at once. | Yes |
Proposition 1.107. The poset \( \sum \) is a simplicial complex. | Proof. We may assume that \( \left( {W, V}\right) \) is essential, since \( \sum \) remains unchanged, up to canonical isomorphism, if we pass to the essential part. According to Definition A.1, we must check two conditions. Condition (a) is that any two elements of \( \sum \) have a greatest lower bound; this has alre... | Yes |
Proposition 1.108. The geometric realization \( \left| \sum \right| \) is canonically homeomorphic to a sphere of dimension \( \operatorname{rank}\left( {W, V}\right) - 1 \) . | Proof. Again we may assume that \( \left( {W, V}\right) \) is essential, in which case we will exhibit a homeomorphism from \( \left| \sum \right| \) to the unit sphere in \( V \) . Recall from Section A.1.1 that \( \left| \sum \right| \) consists of certain convex combinations \( \mathop{\sum }\limits_{v}{\lambda }_{v... | Yes |
Proposition 1.113. Let \( T \) be the set of reflections in \( W \) . Then \( T \) is the set of conjugates of elements of \( S \), and there is a bijection \( \mathcal{H} \rightarrow T \) given by \( H \mapsto {s}_{H} \) . This bijection is \( W \) -equivariant, where \( W \) acts on \( T \) by conjugation. | Proof. The fact that \( T \) is the set of conjugates of elements of \( S \) was proved in step (b) of the proof of Theorem 1.69, and the bijection with \( \mathcal{H} \) follows from Corollary 1.72. Finally, \( W \) -equivariance is simply the familiar formula \( {s}_{wH} = w{s}_{H}{w}^{-1} \), which we have already u... | Yes |
Lemma 1.115. Let \( \alpha \) be a positive root with corresponding hyperplane \( H \mathrel{\text{:=}} \) \( {\alpha }^{ \bot } \) . For any \( w \in W \) , \( {w\alpha } \) is a negative root if and only if \( H \) separates \( C \) from \( {w}^{-1}C \) . | Proof. Let \( U \) be the half-space corresponding to \( \alpha \) . Then\n\n\[ \n{w\alpha }\text{is negative} \Leftrightarrow {wU}\text{is negative} \n\]\n\n\[ \n\Leftrightarrow {wU}\text{does not contain}C \n\]\n\n\[ \n\Leftrightarrow U\text{does not contain}{w}^{-1}C \n\]\n\n\[ \n\Leftrightarrow H\text{separates}C\t... | Yes |
Proposition 1.118. For all \( s \in S \) and \( w \in W,{wC} \subseteq {U}_{ + }\left( s\right) \) if and only if \( l\left( {sw}\right) > l\left( w\right) \) . | Proof. We have\n\n\[ \n{wC} \subseteq {U}_{ + }\left( s\right) \Leftrightarrow {H}_{s}\text{ does not separate }C\text{ from }{wC} \n\]\n\n\[ \n\Leftrightarrow d\left( {{sC},{wC}}\right) > d\left( {C,{wC}}\right) \n\]\n\n\[ \n\Leftrightarrow d\left( {C,{swC}}\right) > d\left( {C,{wC}}\right) \n\]\n\n\[ \n\Leftrightarro... | Yes |
We illustrate the concepts in this section by applying them to the case that \( W \) is the symmetric group on \( n \) letters acting on \( {\mathbb{R}}^{n} \) as in Examples 1.10 and 1.81. Recall that a permutation \( \pi \) acts on \( {\mathbb{R}}^{n} \) by \( \pi {e}_{i} = {e}_{\pi \left( i\right) } \) , where \( {e... | Let's take the analysis one step further and identify the roots with subsets of \( W \) . Here if \( \alpha \) is a root and \( U \) is the associated half-space \( \langle \alpha , - \rangle > 0 \), the corresponding subset of \( W \) is \( \{ w \in W \mid {wC} \subseteq U\} \) . Thinking of elements of \( W \) as per... | Yes |
Proposition 1.127. The simplicial complex \( \sum \) associated to a finite reflection group is a flag complex. | Proof. In view of the characterization of flag complexes given in Proposition A.7, it suffices to note that every set of pairwise joinable simplices is joinable. This is in fact true in greater generality. Indeed, if \( \sum \) is the poset of cells associated to an arbitrary hyperplane arrangement, then every set of p... | No |
Proposition 1.128. The chamber complex \( \sum \) associated with a finite reflection group \( W \) is colorable. | Proof. We will use the criterion in terms of retractions stated at the end of Section A.1.3. Choose a chamber \( C \) . Then we can define \( \phi : \sum \rightarrow {\sum }_{ \leq C} \) by letting \( \phi \left( A\right) \) be the unique face of \( C \) that is \( W \) -equivalent to \( A \) (see Theorem 1.104). It is... | Yes |
Corollary 1.129. The action of \( W \) on \( \sum \) is type-preserving. | Recall from Section A.1.3 that the (essentially unique) type function on \( \sum \) ’ is completely determined once one assigns types to the vertices of a \ | No |
Proposition 1.130. The type-change map \( {\left( {\mathrm{{op}}}_{\sum }\right) }_{ * } \) is \( {\sigma }_{0} \) . | Proof. Fix \( s \in S \) and let \( A \) be the panel of \( C \) of cotype \( s \), i.e., the panel of \( C \) fixed by \( s \) . We have to show that the cotype of the panel \( - A \) of \( - C \) is \( {\sigma }_{0}\left( s\right) \) . In view of Corollary 1.129, the cotype of \( - A \) is the same as that of \( {w}_... | Yes |
Corollary 1.131. If \( \\left( {W, V}\\right) \) is essential, then the following conditions are equivalent:\n\n(i) The involution \( {\\sigma }_{0} \) is trivial.\n\n(ii) \( {w}_{0} \) is central in \( W \) .\n\n(iii) The opposition involution of \( \\sum = \\sum \\left( {W, V}\\right) \) is given by the action of \( ... | Proof. From the original definition of \( {\\sigma }_{0} \), we see that it is trivial if and only if \( {w}_{0} \) commutes with each \( s \\in S \) . Hence (i) and (ii) are equivalent. On the other hand, Proposition 1.130 shows that (i) holds if and only if \( {\\mathrm{{op}}}_{\\sum } \) is type-preserving. But this... | Yes |
Proposition 1.135. Let \( \sum \) be the poset of cells associated to a hyperplane arrangement. For any \( A \in \sum \) and any chambers \( C, D \in {\sum }_{ \geq A} \), every minimal gallery joining \( C \) to \( D \) lies entirely in \( {\sum }_{ \geq A} \) . In particular, if \( \sum \) is simplicial (and hence a ... | Proof. This has already been proved in Exercise 1.67, but here is an independent proof. Given chambers \( C, D \), let \( \Gamma : C = {C}_{0},\ldots ,{C}_{l} = D \) be a minimal gallery. Then the walls \( {H}_{1},\ldots ,{H}_{l} \) crossed by \( \Gamma \) separate \( C \) from \( D \) (Proposition 1.56). For each \( i... | Yes |
Lemma 2.2. Suppose \( \left( {W, S}\right) \) satisfies (A). Then one can associate to each \( w \in W \) a finite subset \( T\left( w\right) \subseteq T \) with the following properties:\n\n(1) \( \left| {T\left( w\right) }\right| = l\left( w\right) \), where \( l \mathrel{\text{:=}} {l}_{S} \) is the length function ... | Proof. It should be clear that heuristically, \( T\left( w\right) \) is supposed to be the set of reflections with respect to the \ | No |
Corollary 2.3. If \( \left( {W, S}\right) \) satisfies (A), then it satisfies (D). | Proof. Suppose \( w = {s}_{1}\cdots {s}_{m} \) with \( m > l\left( w\right) \), and consider the corresponding sequence of reflections \( {t}_{i}\left( {i = 1,\ldots, m}\right) \) . These cannot all be distinct, since this would imply, by assertion (3) of Lemma 2.2, \( \left| {T\left( w\right) }\right| = m \), contradi... | Yes |
Proposition 2.13. The function \( J \mapsto {W}_{J} \) is a poset isomorphism from the set of subsets of \( S \) to the set of standard subgroups of \( W \), where both sets are ordered by inclusion. The inverse is given by \( {W}^{\prime } \mapsto {W}^{\prime } \cap S \) . | Proof. Consider the map from standard subgroups to subsets of \( S \) given by \( {W}^{\prime } \mapsto {W}^{\prime } \cap S \) for any standard subgroup \( {W}^{\prime } \) . It is clear that \( {W}^{\prime } = \left\langle {{W}^{\prime } \cap S}\right\rangle \) if \( {W}^{\prime } \) is a standard subgroup. It is als... | Yes |
Proposition 2.14. Let \( {W}_{J} \) be a standard subgroup, where \( J \subseteq S \) . For any \( w \in {W}_{J} \) | Proof. Suppose we have a \( J \) -reduced decomposition\n\n\[ w = {s}_{1}\cdots {s}_{d} \]\n\n(2.2)\n\nThus \( {s}_{i} \in J \) for each \( i \) and there is no shorter \( J \) -word representing \( w \) . We must show that there is no shorter \( S \) -word representing \( w \) . If there were a shorter \( S \) -word r... | No |
Lemma 2.15. Given \( J \subseteq S, w \in {W}_{J} \), and \( s \in S \smallsetminus J \), we have \( l\left( {sw}\right) = \) \( l\left( w\right) + 1 \) . | Proof. Choose a reduced decomposition \( w = {s}_{1}\cdots {s}_{l} \) with \( {s}_{i} \in J \) for all \( i \) . Suppose \( l\left( {sw}\right) < l\left( w\right) \) . Then \( w = s{s}_{1}\cdots {\widehat{s}}_{i}\cdots {s}_{l} \) for some \( i \) by the exchange condition. This implies that \( s \in {W}_{J} \cap S = J ... | Yes |
Proposition 2.16. For any \( w \in W \) there is a subset \( S\left( w\right) \subseteq S \) such that all reduced decompositions of \( w \) involve precisely the letters in \( S\left( w\right) \) . Moreover, \( S\left( w\right) \) is the smallest subset \( J \subseteq S \) with \( w \in {W}_{J} \) . | Proof. Both assertions will follow if we prove the following: Given two decompositions \( w = {s}_{1}\cdots {s}_{l} = {t}_{1}\cdots {t}_{r} \) with the one on the left reduced, each \( {s}_{i} \) is equal to some \( {t}_{j} \) . We argue by induction on \( l = l\left( w\right) \), which may be assumed \( > 0 \) . Let \... | Yes |
Proposition 2.17. Fix \( w \in W \) and let \( J \mathrel{\text{:=}} \{ s \in S \mid l\left( {sw}\right) < l\left( w\right) \} \) . Then every reduced \( J \) -word can occur as an initial subword of a reduced decomposition of \( w \) . Hence\n\n\[ l\left( {{w}^{\prime }w}\right) = l\left( w\right) - l\left( {w}^{\prim... | Proof. Let \( {t}_{1}\cdots {t}_{l} \) be a reduced \( J \) -word. Arguing by induction on \( l \), we may assume that we have a reduced decomposition\n\n\[ w = {t}_{2}\cdots {t}_{l}{s}_{1}\cdots {s}_{r} \]\n\nSince \( l\left( {{t}_{1}w}\right) < l\left( w\right) \), we can exchange one of the letters in this decomposi... | Yes |
Corollary 2.19. \( W \) is finite if and only if it has an element \( {w}_{0} \) such that \( l\left( {s{w}_{0}}\right) \leq l\left( {w}_{0}\right) \) for all \( s \in S \) . In this case \( {w}_{0} \) has maximal length and is the unique element of maximal length, and it has order 2. Moreover, \[ l\left( {w{w}_{0}}\ri... | Proof. If \( W \) is finite, it obviously has an element \( {w}_{0} \) of maximal length, and then necessarily \( l\left( {s{w}_{0}}\right) \leq l\left( {w}_{0}\right) \) for all \( s \in S \) . Conversely, if \( {w}_{0} \) is an element such that \( l\left( {s{w}_{0}}\right) \leq l\left( {w}_{0}\right) \) for all \( s... | Yes |
Proposition 2.20. Let \( {W}_{J} \) be a standard subgroup \( \left( {J \subseteq S}\right) \) . Then every left coset \( w{W}_{J} \) has a unique representative \( {w}_{1} \) of minimal length. It is characterized by the property\n\n\[ l\left( {{w}_{1}s}\right) = l\left( {w}_{1}\right) + 1 \] \n\n(2.5) \n\nfor all \( ... | Proof. Choose \( {w}_{1} \) of minimal length in the coset. Then \( l\left( {{w}_{1}s}\right) \geq l\left( {w}_{1}\right) \) for all \( s \in J \) ; hence (2.5) holds. To prove (2.6) (which implies the uniqueness of \( {w}_{1} \) ), choose a reduced decomposition \( {w}_{1} = {s}_{1}\cdots {s}_{l} \), and consider an a... | Yes |
Proposition 2.23. Let \( {W}_{J} \) and \( {W}_{K} \) be standard subgroups \( \left( {J, K \subseteq S}\right) \) . Then every \( \left( {{W}_{J},{W}_{K}}\right) \) -double coset \( {W}_{J}w{W}_{K} \) has a unique representative \( {w}_{1} \) of minimal length. It is \( \left( {J, K}\right) \) -reduced and is the uniq... | The proof will use the following consequence of (D):\n\nLemma 2.24. Let \( J \) | No |
Lemma 2.24. Let \( J \) and \( K \) be subsets of \( S \), and let \( {w}_{1} \) be an element of minimal length in its double coset \( {W}_{J}{w}_{1}{W}_{K} \) . Suppose \( u \in {W}_{J} \) and \( v \in {W}_{K} \) are elements such that \( l\left( {u{w}_{1}v}\right) < l\left( u\right) + l\left( {w}_{1}\right) + l\left... | Proof. Consider the decomposition of \( u{w}_{1}v \) obtained by combining the given decompositions of \( u \) and \( v \) with a reduced decomposition of \( {w}_{1} \) . By hypothesis this is not reduced, so we can delete two letters. The assumption on \( {w}_{1} \) implies that neither of the deleted letters can invo... | Yes |
Lemma 2.25. With the notation of Proposition 2.23,\n\n\\[ \n{W}_{J} \cap {w}_{1}{W}_{K}{w}_{1}^{-1} = {W}_{{J}_{1}} \n\\]\n\nwhere \\( {J}_{1} \\mathrel{\\text{:=}} J \cap {w}_{1}K{w}_{1}^{-1} \\) . | Proof. Given \\( u \in {W}_{J} \cap {w}_{1}{W}_{K}{w}_{1}^{-1} \\), we must show that \\( u \in {W}_{{J}_{1}} \\) . Equivalently, given \\( u \in {W}_{J} \\) and \\( v \in {W}_{K} \\) such that \\( u{w}_{1}v = {w}_{1} \\), we must show that \\( u \in {W}_{{J}_{1}} \\) . Note first that by repeated applications of Lemma... | Yes |
Corollary 2.36. \( W \) admits the presentation \[ W = \left\langle {S;{\left( st\right) }^{m\left( {s, t}\right) } = 1}\right\rangle , \] where there is one relation for each pair \( s, t \) with \( m\left( {s, t}\right) < \infty \) . | Proof. Let \( \widetilde{W} \) be the abstract group defined by this presentation, and consider the canonical surjection \( \widetilde{W} \rightarrow W \) . By Theorem 2.33, an element \( \widetilde{w} \) in the kernel can be represented by a word \( \mathbf{s} \) that is reducible to the empty word by \( M \) -operati... | No |
Lemma 2.37. If \( w \in W \) and \( s \in S \smallsetminus S\left( w\right) \) satisfy \( l\left( {sws}\right) < l\left( w\right) + 2 \), then \( s \) commutes with all elements of \( S\left( w\right) \) . | Proof. We have \( l\left( {sw}\right) = l\left( w\right) + 1 = l\left( {ws}\right) \) by Lemma 2.15. Therefore, in view of the folding condition (Section 2.3.1), the hypothesis \( l\left( {sws}\right) < l\left( w\right) + 2 \) is equivalent to the equation \( {sw} = {ws} \) . We now show by induction on \( l \mathrel{\... | Yes |
Lemma 2.44. Given \( s, t \in S \) with \( m\left( {s, t}\right) > 2 \), let \( J \mathrel{\text{:=}} S \smallsetminus \{ s\} \) and \( K \mathrel{\text{:=}} \) \( J \smallsetminus \{ t\} = S \smallsetminus \{ s, t\} \) . Suppose \( w \) is a right \( K \) -reduced element of \( {W}_{J} \) . Then ws is (right) J-reduce... | Proof. Note first that \( l\left( {ws}\right) = l\left( w\right) + 1 \) by Lemma 2.15. We must show that \( l\left( {wsr}\right) = l\left( w\right) + 2 \) for \( r \in S \smallsetminus \{ s\} \) . If \( r \neq t \), then \( r \in K \) and \( w \) is \( r \) -reduced. Since \( w \) is also \( s \) -reduced, we have \( l... | Yes |
Proposition 2.45. Assume that \( \left( {W, S}\right) \) is irreducible, \( S \) is finite, and \( W \) is infinite. If \( I \) and \( J \) are proper subsets of \( S \), then \( {W}_{I} \smallsetminus W/{W}_{J} \) is infinite. | Proof of Proposition 2.45. We argue by induction on \( \left| S\right| \) . We may assume that \( I = S \smallsetminus \left\{ {s}^{\prime }\right\} \) and \( J = S \smallsetminus \{ s\} \) for some \( {s}^{\prime }, s \in S \) . We may also assume, in view of Proposition 2.43, that \( {W}_{I} \) and \( {W}_{J} \) are ... | Yes |
Lemma 2.58. Fix \( s \in S \) and let \( {U}_{ + }\left( s\right) \) and \( {U}_{ - }\left( s\right) \) be the open half-spaces in \( {V}^{ * } \) defined, respectively, by \( \left\langle {-,{e}_{s}}\right\rangle > 0 \) and \( \left\langle {-,{e}_{s}}\right\rangle < 0 \) . Then for any \( w \in W \) we have\n\n\[ \n{w... | Proof. We argue by induction on \( l\left( w\right) \) . If \( l\left( {sw}\right) < l\left( w\right) \), then we may apply the induction hypothesis to the element \( {sw} \) to get \( {swC} \subseteq {U}_{ + }\left( s\right) \) ; multiplying by \( s \) , we obtain \( {wC} \subseteq s{U}_{ + }\left( s\right) = {U}_{ - ... | Yes |
Lemma 2.62. For any \( s \in S,{H}_{s} \) is the only wall separating \( C \) from \( {sC} \) . | Proof. We will use the theory of (finite) hyperplane arrangements developed in Section 1.4. Let \( {\mathcal{H}}_{0} \subseteq \mathcal{H} \) be any finite subset containing the walls of \( C \) and \( {sC} \) . Then \( C \) and \( {sC} \) are \( {\mathcal{H}}_{0} \) -chambers with a common wall \( {H}_{s} = s{H}_{s} \... | Yes |
Theorem 2.65. If \( W \) is finite, then the bilinear form \( B \) is positive definite and \( W \) is a finite reflection group acting on \( V \), with \( S \) as the set of reflections with respect to the chamber defined by \( B\left( {{e}_{s}, - }\right) > 0 \) for all \( s \in S \) . | Proof. Let \( {V}_{1} = {V}^{ * } \), and let’s temporarily forget that \( {V}_{1} \) is the dual of \( V \) . By Theorem 2.59, \( W \) can be identified with a finite group of linear transformations of \( {V}_{1} \) generated by linear reflections. We can make them orthogonal reflections by putting a \( W \) -invarian... | No |
Proposition 2.67. If \( B \) is positive definite, then \( W \) is finite. | Proof. \( W \) is a subgroup of the orthogonal group consisting of all linear transformations of \( V \) that leave \( B \) invariant. Since \( B \) is positive definite, this orthogonal group is compact. In view of Theorem 2.59, \( W \) is a discrete subgroup of a compact group; hence it is finite. | Yes |
Proposition 2.69. The following conditions on a Coxeter system \( \left( {W, S}\right) \) are equivalent:\n\n(i) \( W \) is finite.\n\n(ii) \( - C \) is a chamber.\n\n(iii) \( \mathcal{H} \) is finite.\n\n(iv) \( T \) is finite.\n\n(v) \( \Phi \) is finite.\n\n(vi) \( \mathcal{C} \) is finite. | Proof. It is clear from the discussion in Section 2.5.3 that conditions (iii), (iv), and (v) are all equivalent. So it suffices to show\n\n\[ \text{(i)} \Rightarrow \text{(ii)} \Rightarrow \text{(iii)} \Rightarrow \text{(vi)} \Rightarrow \text{(i).} \]\n\nIf \( W \) is finite, then it is a finite reflection group, so \... | Yes |
Proposition 2.73. If \( \\left( {W, S}\\right) \) is an irreducible Coxeter system with \( W \) infinite, then the normalizer of \( S \) is trivial. In particular, the center of \( W \) is trivial. | Proof. The proof is almost the same as the algebraic proof of Corollary 1.91, except that the bilinear form \( B\\left( {-, - }\\right) \) replaces the inner product \( \\langle - , - \\rangle \), and one has to work in both \( V \), which contains the set \( \\Phi \) of roots, and \( {V}^{ * } \), which contains the c... | Yes |
Lemma 2.76. Let \( {W}^{\prime } \leq W \) be a subgroup generated by two reflections. If \( {W}^{\prime } \) is finite, then \( {W}^{\prime } \) is contained in a finite parabolic subgroup. | This is a special case of Proposition 2.87, which we will prove in the next section using the Tits cone. Alternatively, there is a direct combinatorial proof of this special case that will arise naturally in our study of Coxeter complexes in the next chapter; see Corollary 3.167. | No |
Lemma 2.77. Let \( \alpha \) and \( \beta \) be roots with \( \alpha \neq \pm \beta \), and let \( {s}_{\alpha } \) and \( {s}_{\beta } \) be the corresponding reflections. Then \( {s}_{\alpha }{s}_{\beta } \) has finite order if and only if \( \left| {B\left( {\alpha ,\beta }\right) }\right| < 1 \) . Moreover, the set... | Proof. Let \( {W}^{\prime } \) be the dihedral group generated by \( {s}_{\alpha } \) and \( {s}_{\beta } \) ; it is finite if and only if \( {s}_{\alpha }{s}_{\beta } \) has finite order. Suppose first that \( \left| {B\left( {\alpha ,\beta }\right) }\right| < 1 \), and let \( {V}^{\prime } \) be the 2-dimensional sub... | Yes |
Theorem 2.80. The cone \( X \) is convex. For any \( x, y \in X \), the line segment \( \left\lbrack {x, y}\right\rbrack \) crosses only finitely many walls and is contained in a finite union of cells. Moreover: | Proof. To prove the first part of the theorem, we may assume \( x \in \bar{C} \) and \( y \in w\bar{C} \) for some \( w \in W \) . Then \( \left\lbrack {x, y}\right\rbrack \) crosses only finitely many walls by Lemma 2.63, since any wall that it crosses separates \( C \) from \( {wC} \) . We will prove by induction on ... | Yes |
Lemma 2.84. Let \( {\sum }_{ \geq A} \) be the set of cells of \( X \) having \( A \) as a face, and let \( {\sum }_{A} \) be the set of \( {\mathcal{H}}_{A} \) -cells that meet \( X \) . For any cell \( B \in {\sum }_{ \geq A} \), let \( f\left( B\right) \) be the \( {\mathcal{H}}_{A} \) -cell containing \( B \) . The... | Proof. On the level of sign sequences, \( f \) just picks out the components of \( \sigma \left( B\right) \) corresponding to the hyperplanes in \( {\mathcal{H}}_{A} \) . It is \( 1 - 1 \) because the remaining components of \( \sigma \left( B\right) \) are the same as those of \( \sigma \left( A\right) \) . To prove t... | Yes |
Lemma 2.85. Let \( {W}_{A} \) be the stabilizer of \( A \) . Then \( {W}_{A} \) is finite if and only if \( {\mathcal{H}}_{A} \) is finite. | Proof. If \( {W}_{A} \) is finite, then it contains only finitely many reflections, so \( {\mathcal{H}}_{A} \) is finite. Conversely, if \( {\mathcal{H}}_{A} \) is finite, then there are only finitely many \( {\mathcal{H}}_{A} \) - cells, and hence \( {\sum }_{A} \) is finite. Lemma 2.84 now implies that \( {\sum }_{ \... | Yes |
Lemma 2.86. Let \( {X}_{f} \) be the set of points \( x \in X \) whose stabilizer \( {W}_{x} \) is finite. Given \( x \in {X}_{f} \) and \( y \in X \) with \( x \neq y \), the half-open line segment \( \lbrack x, y) \) is contained in \( {X}_{f} \) . In particular, \( {X}_{f} \) is convex. | Proof. In view of Lemma 2.85, \( {X}_{f} \) consists of the points \( x \in X \) such that \( x \) is contained in only finitely many walls. The result now follows from the fact that \( \left\lbrack {x, y}\right\rbrack \) crosses only finitely many walls. | No |
Proposition 2.87. Every finite subgroup of \( W \) is contained in a finite parabolic subgroup. | Proof. Note that \( {X}_{f} \) is \( W \) -invariant and that by Theorem 2.80, the stabilizers \( {W}_{x} \) for \( x \in {X}_{f} \) are precisely the finite parabolic subgroups of \( W \) . So our task is to show that every finite subgroup \( {W}^{\prime } \) of \( W \) fixes a point of \( {X}_{f} \) . The latter bein... | Yes |
Lemma 2.92. Suppose \( W \) is infinite and irreducible. For any \( x \neq 0 \) in \( X \) , there are infinitely many walls not containing \( x \) . | Proof. We may assume that the cell \( A \) containing \( x \) is a face of the fundamental chamber \( C \) and hence that its stabilizer is \( {W}_{J} \) for some \( J \subsetneqq S \) . Suppose \( x \) (and hence \( A \) ) is contained in all but finitely many walls. Then there is an upper bound on the gallery distanc... | Yes |
Lemma 2.93. Suppose that axiom (H3) holds. Given \( A \in \sum \) and \( {\mathcal{H}}_{A} \) as in (H3), the \( {\mathcal{H}}_{A} \) -faces of \( A \) are the same as the \( \mathcal{H} \) -faces of \( A \) . | Proof. Since \( {\mathcal{H}}_{A} \subseteq \mathcal{H} \), the partition of \( \bar{A} \) into \( \mathcal{H} \) -cells refines the partition into \( {\mathcal{H}}_{A} \) -cells. It therefore suffices to show that every \( {\mathcal{H}}_{A} \) -face of \( A \) is contained in an \( \mathcal{H} \) -cell. Let \( B \) be... | Yes |
Proposition 2.94. In the presence of (H1) and (H2), axioms (H4) and (H5) are equivalent to one another. When these axioms are satisfied, the product of cells can be characterized as follows: Given \( x \in A \) and \( y \in B \), the cell \( {AB} \) contains \( \left( {1 - t}\right) x + {ty} \in {AB} \) for all suffici... | Proof. Suppose (H4) holds. Given \( x, y \in X \), let \( A \) (resp. \( B \) ) be the cell containing \( x \) (resp. \( y \) ). We show by induction on \( \left| {\mathcal{S}\left( {A, B}\right) }\right| \) that the open segment \( \left( {x, y}\right) \) is contained in \( X \) . If \( \mathcal{S}\left( {A, B}\right)... | Yes |
Theorem 3.5. The poset \( \sum \mathrel{\text{:=}} \sum \left( {W, S}\right) \) is a simplicial complex. Moreover, it is a thin chamber complex of rank equal to \( \left| S\right| \), it is colorable, and the action of \( W \) on \( \sum \) is type-preserving. | Proof. To show that \( \sum \) is simplicial, there are two things we must verify (see Definition A.1):\n\n(a) Any two elements \( A, B \in \sum \) have a greatest lower bound.\n\n(b) For any \( A \in \sum \), the poset \( {\sum }_{ \leq A} \) is a Boolean lattice.\n\nFor (a) we can use the \( W \) -action on \( \sum \... | Yes |
Example 3.7. Let \( W \) be the group of isometries of the plane generated by the (affine) reflections with respect to the sides of an equilateral triangle. This is an example of a Euclidean reflection group. Although we will not treat the theory of such groups systematically until Chapter 10, the reader should find it... | We will give an ad hoc proof of this in Section 3.4.2 below (Example 3.76); in the meantime, the reader is advised to take the assertion on faith. | No |
Proposition 3.16. Given \( A \in \sum \), let \( J \mathrel{\text{:=}} S \smallsetminus \tau \left( A\right) \) be its cotype. Then \( {\operatorname{lk}}_{\sum }A \) is isomorphic to the Coxeter complex \( \sum \left( {{W}_{J}, J}\right) \) associated to the Coxeter system \( \left( {{W}_{J}, J}\right) \) . In particu... | Proof. We may assume that \( A \) is a face of the fundamental chamber. Then \( A \) \( {is} \) the standard subgroup \( {W}_{J} \) . Recall now that there is a poset isomorphism \( {\operatorname{lk}}_{\sum }A \cong {\sum }_{ \geq A} \) ; hence the link of \( A \) is isomorphic to the set of standard cosets in \( W \)... | Yes |
Corollary 3.17. \( \sum \) is completely determined by its underlying chamber system. More precisely, the simplices of \( \sum \) are in \( 1 - 1 \) correspondence with the residues in \( \mathcal{C}\left( \sum \right) \), ordered by reverse inclusion. Here a simplex \( A \) corresponds to the residue \( \mathcal{C}{\l... | Remark 3.18. We have included this corollary only to force the reader to learn the terminology associated with chamber systems, especially the concept of residue. But the statement of the corollary is in fact a complete tautology in view of the definition of \( \sum \) in terms of standard cosets. Indeed, if one identi... | No |
Corollary 3.20. The Coxeter matrix \( M \) of \( \left( {W, S}\right) \) can be recovered from \( \sum \) ’ as follows: For any \( s, t \in S \) with \( s \neq t, m\left( {s, t}\right) \) is the unique number \( m \) \( \left( {2 \leq m \leq \infty }\right) \) such that the link of every simplex of cotype \( \{ s, t\} ... | This shows, in particular, that the Coxeter group \( W \) is determined up to isomorphism by \( \sum \) . We will see this again in the next section, from a different point of view. | No |
Corollary 3.27. The following conditions are equivalent:\n\n(i) \( \sum \) is a manifold.\n\n(ii) \( \sum \) is locally finite.\n\n(iii) Every proper standard subgroup of \( W \) is finite. | For example, the Coxeter complex associated to \( {\mathrm{{PGL}}}_{2}\left( \mathbb{Z}\right) \) is not a manifold. One can see the nonmanifold points in Figures 2.3 and 2.5: They are the cusps. | No |
Lemma 3.31. Endomorphisms \( \phi \) of \( \sum \) are in \( 1 - 1 \) correspondence with pairs \( \left( {{\phi }^{\prime },{\phi }^{\prime \prime }}\right) \), where \( {\phi }^{\prime } \) is a function \( W \rightarrow W,{\phi }^{\prime \prime } \) is a permutation of \( S \), and \( {\phi }^{\prime }\left( {ws}\ri... | Proof. Let \( \phi \) be an endomorphism of \( \sum \) . Then the restriction of \( \phi \) to the chambers yields a function \( {\phi }^{\prime } : W \rightarrow W \) . (Recall that the chambers are the singleton standard cosets and are identified with the elements of \( W \) .) We also have a type-change map \( {\phi... | Yes |
Proposition 3.32. The image of \( W \hookrightarrow \) Aut \( \sum \) is the normal subgroup \( {\operatorname{Aut}}_{0}\sum \) consisting of the type-preserving automorphisms of \( \sum \) . | Proof. We already know that \( W \) acts as a group of type-preserving automor-phisms of \( \sum \) . Conversely, suppose \( \phi \) is an arbitrary type-preserving automorphism, and let \( {\phi }^{\prime } \) and \( {\phi }^{\prime \prime } \) be its \ | No |
Proposition 3.33. The homomorphism \( \operatorname{Aut}\left( {W, S}\right) \rightarrow \operatorname{Aut}\sum \) just defined is injective, and its image is the group \( \operatorname{Aut}\left( {\sum, C}\right) \) consisting of the automor-phisms of \( \sum \) that stabilize the fundamental chamber \( C = 1 \) . | Proof. Given \( f \in \operatorname{Aut}\left( {W, S}\right) \), its image \( \phi \in \operatorname{Aut}\sum \) has components \( {\phi }^{\prime } = f \) and \( {\phi }^{\prime \prime } \mathrel{\text{:=}} {\left. f\right| }_{S} \) . This shows that the homomorphism is injective. And \( \phi \) stabilizes \( C \) bec... | Yes |
Proposition 3.38. Let \( {C}_{1} \) and \( {C}_{2} \) be adjacent chambers of \( \sum = \sum \left( {W, S}\right) \) . Then there is an endomorphism \( \phi \) of \( \sum \) with the following properties:\n\n\( \left( 1\right) \phi \) is a retraction onto its image \( \alpha \) .\n\n(2) Every chamber in \( \alpha \) is... | To construct \( \phi \), we may assume that \( {C}_{1} \) is the fundamental chamber \( C \) , in which case \( {C}_{2} \) is necessarily \( {sC} \) for some \( s \in S \) . Before beginning the proof based on Lemma 3.31, we remark that there is a very short proof that uses the Tits cone instead of the proposition. Nam... | Yes |
Lemma 3.39. Fix \( s \in S \) . Then there is a function \( {\phi }_{s} : W \rightarrow W \) with the following properties:\n\n(1) \( \phi \) is a retraction onto its image \( {\alpha }_{s} \), which consists of the elements \( w \in W \) such that \( l\left( {sw}\right) = l\left( w\right) + 1 \) .\n\n(2) Each element ... | Proof. It is clear how we should define \( {\phi }_{s} \) :\n\n\[ \n{\phi }_{s}\left( w\right) \mathrel{\text{:=}} \begin{cases} w & \text{ if }l\left( {sw}\right) = l\left( w\right) + 1 \\ {sw} & \text{ if }l\left( {sw}\right) = l\left( w\right) - 1 \end{cases} \n\]\n\nAnd it is immediate from this definition that (1)... | Yes |
Lemma 3.42. \( {\phi }^{\prime } \) takes adjacent chambers to chambers that are equal or adjacent. | Proof. Let \( C \) and \( D \) be adjacent chambers. If they are both in \( {\alpha }^{\prime } \), there is nothing to prove. So assume that at least one of them, say \( C \), is in \( \alpha \) . Then \( {\phi }^{\prime }\left( C\right) \) is the unique chamber \( {C}^{\prime } \in {\alpha }^{\prime } \) such that \(... | No |
Lemma 3.43. There exists a pair \( C,{C}^{\prime } \) of adjacent chambers with \( C \in \alpha \) and \( {C}^{\prime } \in {\alpha }^{\prime } \) . For any such pair, we have \( \phi \left( {C}^{\prime }\right) = C \) and \( {\phi }^{\prime }\left( C\right) = {C}^{\prime } \) . | Proof. Since \( \mathcal{C}\left( \alpha \right) \) and \( \mathcal{C}\left( {\alpha }^{\prime }\right) \) are both nonempty, there is a gallery \( \Gamma \) that starts in \( \alpha \) and ends in \( {\alpha }^{\prime } \) . Then \( \Gamma \) must cross from \( \alpha \) to \( {\alpha }^{\prime } \) at some point, whe... | Yes |
Lemma 3.44. \( \alpha \) and \( {\alpha }^{\prime } \) are convex subcomplexes of \( \sum \), in the sense that if \( \Gamma \) is a minimal gallery in \( \sum \) with both extremities in \( \alpha \) (resp. \( {\alpha }^{\prime } \) ), then \( \Gamma \) lies entirely in \( \alpha \) (resp. \( {\alpha }^{\prime } \) ). | Proof. Suppose \( \Gamma \) is a minimal gallery with both extremities in \( \alpha \) . If \( \Gamma \) is not contained in \( \alpha \), then it must cross from \( \alpha \) to \( {\alpha }^{\prime } \) at some point. Thus there is a pair of consecutive chambers in \( \Gamma \) to which we can apply Lemma 3.43. But t... | Yes |
Lemma 3.45. Let \( C \) and \( {C}^{\prime } \) be as in Lemma 3.43. Then\n\n\[ \mathcal{C}\left( \alpha \right) = \left\{ {D \in \mathcal{C}\left( \sum \right) \mid d\left( {D, C}\right) < d\left( {D,{C}^{\prime }}\right) }\right\} \]\n\nand\n\n\[ \mathcal{C}\left( {\alpha }^{\prime }\right) = \left\{ {D \in \mathcal{... | Proof. Note that the right-hand sides of the two equalities to be proved are disjoint sets of chambers. Consequently, since \( \alpha \) and \( {\alpha }^{\prime } \) partition the chambers of \( \sum \), it suffices to prove that the left-hand sides are contained in the righthand sides. Suppose, then, that we are give... | Yes |
Lemma 3.46. Suppose \( C \) and \( {C}^{\prime } \) are adjacent chambers such that \( \phi \left( {C}^{\prime }\right) = C \) . Then \( \phi \) is the unique folding taking \( {C}^{\prime } \) to \( C \) . | Proof. Note first that we have \( C \in \phi \left( \sum \right) = \alpha \) and \( {C}^{\prime } \in {\alpha }^{\prime } \) [because \( \phi \left( {C}^{\prime }\right) \neq \) \( \left. {C}^{\prime }\right\rbrack \) . So Lemma 3.45 is applicable and yields a description of the two halves \( \alpha \) and \( {\alpha }... | Yes |
Lemma 3.49. Let \( C \) and \( {C}^{\prime } \) be adjacent chambers with \( \phi \left( {C}^{\prime }\right) = C \) . Then \( \phi \) is reversible if and only if there exists a folding taking \( C \) to \( {C}^{\prime } \) . In this case there is an automorphism \( s \) of \( \sum \) such that \( {\left. s\right| }_{... | Proof. We have already seen that if \( \phi \) is reversible then the opposite folding \( {\phi }^{\prime } \) takes \( C \) to \( {C}^{\prime } \) . Conversely, suppose there is a folding \( {\phi }_{1} \) such that \( {\phi }_{1}\left( C\right) = {C}^{\prime } \) . Then we can apply Lemma 3.45 to \( {\phi }_{1} \) to... | Yes |
Let \( W \) be the symmetric group on \( n \) letters with its standard generating set \( S = \left\{ {{s}_{1},\ldots ,{s}_{n - 1}}\right\} \), where \( {s}_{i} \) is the transposition that interchanges \( i \) and \( i + 1 \). For any permutation \( \pi \in W \), let \( \iota \left( \pi \right) \) be the number of inv... | \[ \pi \left( i\right) < \pi \left( {i + 1}\right) \Rightarrow \iota \left( {\pi {s}_{i}}\right) = \iota \left( \pi \right) + 1, \] \[ \pi \left( i\right) > \pi \left( {i + 1}\right) \Rightarrow \iota \left( {\pi {s}_{i}}\right) = \iota \left( \pi \right) - 1. \] Indeed, if one identifies a permutation \( \pi \) with t... | Yes |
Lemma 3.53. If \( \alpha \) and \( \beta \) are distinct roots of a spherical Coxeter complex \( \sum \) , then \( \alpha \nsubseteq \beta \) . | Proof. This is obvious from the point of view of Chapter 1, where roots correspond to half-spaces whose bounding hyperplanes pass through the origin. Alternatively, the lemma follows from the fact that \( \alpha \) and \( \beta \) have the same finite number of chambers, equal to half the number of chambers in \( \sum ... | No |
Lemma 3.54. Roots and walls in a thin chamber complex \( \sum \) are full subcom-plexes. | Proof. For roots, this follows from Lemma A.15. For walls the result follows from the fact that a wall is an intersection of two roots. Alternatively, one can use the fact that a wall is the fixed-point set of a reflection. | Yes |
Theorem 3.65. A thin chamber complex \( \sum \) is a Coxeter complex if and only if every pair of adjacent chambers is separated by a wall. | Proof of Theorem 3.65 (start). We have already proven the \ | No |
Lemma 3.66. \( W \) acts transitively on the chambers of \( \sum \) . | Proof. This is identical to the proof given in Chapter 1 for finite reflection groups (Theorem 1.69). | No |
Lemma 3.67. \( \sum \) is colorable. | Proof. Let \( \bar{C} \) be the subcomplex \( {\sum }_{ \leq C} \) . It suffices to show that \( \bar{C} \) is a retract of \( \sum \) . The idea for showing this is to construct a retraction \( \rho \) by folding and folding and folding…, until the whole complex \( \sum \) has been folded up onto \( \bar{C} \) .\n\nTo... | No |
Lemma 3.68. Foldings and reflections are type-preserving; hence all elements of \( W \) are type-preserving. Consequently, \( {wC} \) and wsC are s-adjacent for any \( w \in W \) and \( s \in S \) . | Proof. A folding \( \phi \) fixes at least one chamber pointwise, so the type-change map \( {\phi }_{ * } \) is the identity (see Proposition A.14). This proves that foldings are type-preserving, and everything else follows from this. | Yes |
Lemma 3.69. If \( \Gamma : {C}_{0},\ldots ,{C}_{d} \) is a minimal gallery, then the walls crossed by \( \Gamma \) are distinct and are precisely the walls separating \( {C}_{0} \) from \( {C}_{d} \) . Hence the distance between two chambers is equal to the number of walls separating them. | Proof. Suppose \( H \) is a wall separating \( {C}_{0} \) from \( {C}_{d} \) . Let \( \pm \alpha \) be the corresponding roots, say with \( {C}_{0} \in \alpha \) and \( {C}_{d} \in - \alpha \) . Then there must be some \( i \) with \( 1 \leq i \leq d \) such that \( {C}_{i - 1} \in \alpha \) and \( {C}_{i} \in - \alpha... | Yes |
Lemma 3.70. Let \( \Gamma \) be a gallery of type \( \mathbf{s} = \left( {{s}_{1},\ldots ,{s}_{d}}\right) \) . If \( \Gamma \) is not minimal, then there is a gallery \( {\Gamma }^{\prime } \) with the same extremities as \( \Gamma \) such that \( {\Gamma }^{\prime } \) has type \( {\mathbf{s}}^{\prime } = \left( {{s}_... | Proof. Since \( \Gamma \) is not minimal, Lemma 3.69 implies that the number of walls separating \( {C}_{0} \) from \( {C}_{d} \) is less than \( d \) . Hence the walls crossed by \( \Gamma \) cannot all be distinct; for if a wall is crossed exactly once by \( \Gamma \), then it certainly separates \( {C}_{0} \) from \... | Yes |
Lemma 3.71. The action of \( W \) is simply transitive on the chambers of \( \sum \) . | Proof. We have already noted that the action is transitive. To prove that the stabilizer of \( C \) is trivial, note that if \( {wC} = C \) then \( w \) fixes \( C \) pointwise, since \( w \) is type-preserving. But then \( w = 1 \) by the standard uniqueness argument. | Yes |
Lemma 3.75. The subcomplex \( \bar{C} \mathrel{\text{:=}} {\sum }_{ \leq C} \) is a simplicial fundamental domain for the action of \( W \) on \( \sum \) . Moreover, the stabilizer of the face of \( C \) of cotype \( J \) is the standard subgroup \( {W}_{J} \) of \( W \) . | Proof. The first assertion follows from the transitivity of \( W \) on the chambers, together with the fact that \( W \) is type-preserving. To prove the second, let \( A \) be a face of \( C \) and let \( \tau \left( A\right) = S \smallsetminus J \) . It follows from the definition of \( \tau \) that \( J \) is the se... | Yes |
Let \( \sum \) be the plane tiled by equilateral triangles. It is geometrically evident that we can construct, for any adjacent chambers \( C,{C}^{\prime } \), a folding taking \( {C}^{\prime } \) to \( C \) . So \( \sum \) is indeed a Coxeter complex, as claimed in Example 3.7. | To see that the Coxeter group \( W \) is the one given in that example, one can compute the orders of pairwise products of fundamental reflections, or one can observe that the link of every vertex is a hexagon. | No |
Proposition 3.78. For any two simplices \( A, B \) in a Coxeter complex \( \sum \), we have\n\n\[ d\left( {A, B}\right) = \left| {\mathcal{S}\left( {A, B}\right) }\right| \]\n\ni.e., \( d\left( {A, B}\right) \) is equal to the number of walls \( H \) that strictly separate \( A \) from \( B \) . More precisely, the wal... | Proof. A proof from the point of view of the Tits cone was sketched in Section 2.7. Here is a combinatorial proof: Let \( \Gamma : {C}_{0},\ldots ,{C}_{d} \) be a minimal gallery from \( A \) to \( B \) . Then it is also a minimal gallery from \( {C}_{0} \) to \( {C}_{d} \), so it crosses \( d \) distinct walls, and th... | Yes |
Proposition 3.79. The function \( H \mapsto {H}^{\prime } \mathrel{\text{:=}} H \cap {\sum }^{\prime } \) is a bijection from the set of walls of \( \sum \) containing \( A \) to the set of walls of \( {\sum }^{\prime } \) . Similarly, the function \( \alpha \mapsto {\alpha }^{\prime } \mathrel{\text{:=}} \alpha \cap {... | Proof. It suffices to prove the first assertion. Since a wall of \( {\sum }^{\prime } \) is completely determined by any panel that it contains, we can reformulate the assertion as follows: For any panel \( {P}^{\prime } \) of \( {\sum }^{\prime } \), there is a unique wall \( H \) of \( \sum \) with \( A \in H \) and ... | Yes |
Proposition 3.85. There is a type-preserving isomorphism \( \sum \cong \sum \left( {{W}_{M}, S}\right) \) , where \( \sum \left( {{W}_{M}, S}\right) \) is given its canonical type function with values in \( S \) . | Proof. By definition, there is a simplicial isomorphism \( \phi : \sum \rightarrow \sum \left( {{W}^{\prime },{S}^{\prime }}\right) \) for some Coxeter system \( \left( {{W}^{\prime },{S}^{\prime }}\right) \) . Let \( {\phi }_{ * } : S \rightarrow {S}^{\prime } \) be the induced type-change bijection (Proposition A.14)... | Yes |
Proposition 3.90. Given simplices \( A, B \in \sum \), we have \( B \leq A \) if and only if \( \sigma \left( B\right) \leq \sigma \left( A\right) \) . In particular, \( A = B \) if and only if \( \sigma \left( A\right) = \sigma \left( B\right) \), i.e., a simplex is uniquely determined by its sign sequence. | Proof. If one wants to use the Tits cone, the result is already contained in Section 2.6 (see the paragraph following Definition 2.79). But here is a purely combinatorial proof.\n\nSuppose \( B \leq A \) . Then \( \sigma \left( B\right) \leq \sigma \left( A\right) \), since every root containing \( A \) must contain \(... | No |
Proposition 3.93. For any simplex \( A \in \sum \), the set \( \mathcal{C}{\left( \sum \right) }_{ \geq A} \) of chambers having \( A \) as a face is convex. More concisely, residues are convex. | Proof. Proposition 3.90 implies that \( \mathcal{C}{\left( \sum \right) }_{ \geq A} \) is an intersection of convex sets of the form \( \mathcal{C}\left( \alpha \right) \) for various roots \( \alpha \), one for each wall \( H \) such that \( {\sigma }_{H}\left( A\right) \neq 0 \) . This proves the first assertion, and... | Yes |
Proposition 3.94. Let \( \mathcal{D} \) be a nonempty set of chambers in \( \sum \) . Then \( \mathcal{D} \) is a convex subset of \( \mathcal{C}\left( \sum \right) \) if and only if \( \mathcal{D} \) is an intersection of sets \( \mathcal{C}\left( \alpha \right) \) for some family of roots \( \alpha \) . | Proof. It suffices to prove the \ | No |
Proposition 3.99. A is a maximal simplex of its support. | Proof. If \( A < B \) in \( \sum \), then Proposition 3.90 implies that there is a wall \( H \) with \( {\sigma }_{H}\left( A\right) = 0 \) and \( {\sigma }_{H}\left( B\right) \neq 0 \) ; hence \( B \notin \operatorname{supp}A \) . | Yes |
Proposition 3.100. If \( \sum = \sum \left( {W, S}\right) \) with its natural \( W \) -action, then the stabilizer \( {W}_{A} \) of any simplex \( A \) fixes \( \operatorname{supp}A \) pointwise, i.e., it fixes every simplex of \( \operatorname{supp}A \) . Moreover, \( \operatorname{supp}A \) is the full fixed-point se... | Proof. This is an immediate consequence of the fact that \( {W}_{A} \) is generated by the reflections that fix \( A \), i.e., the reflections \( {s}_{H} \) with \( A \in H \) ; see Exercise 3.62. | No |
Proposition 3.103. Given a simplex \( A \) and a chamber \( C \), there is a (unique) chamber \( {AC} \) such that for any \( H \in \mathcal{H} \), \[ {\sigma }_{H}\left( {AC}\right) = \left\{ \begin{array}{ll} {\sigma }_{H}\left( A\right) & \text{ if }{\sigma }_{H}\left( A\right) \neq 0, \\ {\sigma }_{H}\left( C\right... | Proof. Choose a minimal gallery \( \Gamma : {C}_{0},\ldots ,{C}_{d} = C \) from \( A \) to \( C \). Given \( H \in \mathcal{H} \), if \( {\sigma }_{H}\left( A\right) \neq 0 \), then \( {\sigma }_{H}\left( {C}_{0}\right) = {\sigma }_{H}\left( A\right) \), since \( {C}_{0} \geq A \). If \( {\sigma }_{H}\left( A\right) = ... | Yes |
For any simplex \( A \) and any chambers \( C, D \) with \( D \geq A \) , \[ d\left( {C, D}\right) = d\left( {C,{AC}}\right) + d\left( {{AC}, D}\right) . \] | Proof. Partition the walls \( H \) separating \( C \) from \( D \) into two subsets according to whether or not \( {\sigma }_{H}\left( A\right) = 0 \) . Those with \( {\sigma }_{H}\left( A\right) = 0 \) are precisely the walls separating \( {AC} \) from \( D \), while those with \( {\sigma }_{H}\left( A\right) \neq 0 \... | Yes |
Lemma 3.107. The projection \( \mathcal{C} \rightarrow {\mathcal{C}}_{A} \) takes adjacent chambers to chambers that are equal or adjacent. Consequently, \( {\mathcal{C}}_{A, B} \) is a connected subset of the chamber graph of \( \sum \), i.e., any two elements of \( {\mathcal{C}}_{A, B} \) can be connected by a galler... | Proof. The first assertion is immediate if one calculates projections in terms of sign sequences (equation (3.10)). The second assertion now follows from part (1) of Proposition 3.106 because \( {\mathcal{C}}_{B} \) is connected (by Proposition 3.16 or Proposition 3.93). | Yes |
Let \( C \) and \( {C}^{\prime } \) be adjacent chambers. Let \( v,{v}^{\prime } \) be the vertices of \( C,{C}^{\prime } \) that are not in the common panel, as in Figure 3.9. Consider the product \( v{v}^{\prime } \). We show by two different methods that \( v{v}^{\prime } \leq C \). | Method 1: There is a minimal gallery \( C,{C}^{\prime } \) from \( v \) to \( {v}^{\prime } \) since \( v \) and \( {v}^{\prime } \) are not joinable. [They have the same type.] Hence \( v{v}^{\prime } \) is a face of the starting chamber \( C \). Method 2: Use sign sequences. Assume for simplicity (and without loss of... | Yes |
Corollary 3.113. For any simplices \( A, B \in \sum ,\dim {AB} = \dim {BA} \) . Consequently, \( \dim {AB} \geq \max \{ \dim A,\dim B\} \) . | Proof. The first assertion follows immediately from parts (5) and (4) of the proposition. For the second, we have \( \dim {AB} \geq \dim A \) trivially because \( A \leq \) \( {AB} \), and similarly \( \dim {BA} \geq \dim B \) ; now use the fact that \( \dim {BA} = \dim {AB} \) . | Yes |
Lemma 3.123. Let \( \sum = \sum \left( {W, S}\right) \), and let \( {W}_{A} \) for \( A \in \sum \) be the stabilizer of \( A \) in \( W \). Then for any two simplices \( A, B \in \sum \) we have \( {W}_{AB} = {W}_{A} \cap {W}_{B} \). | Proof. It is clear that \( {W}_{A} \cap {W}_{B} \leq {W}_{AB} \) and that \( {W}_{AB} \leq {W}_{A} \). [For the latter, note that \( A \leq {AB} \) and \( W \) is type-preserving.] So all that remains to show is that \( {W}_{AB} \) fixes \( B \). This follows from Proposition 3.100 because \( B \in \operatorname{supp}{... | No |
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