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Proposition 3.4.2 Let \( w \in W \) . Then, \( w \) can be uniquely written as \( w = \) \( {x}_{1}\cdots {x}_{n} \), where \( {x}_{i} \in {}^{\left\lbrack i - 1\right\rbrack }\left( {W}_{\left\lbrack i\right\rbrack }\right) \) for \( i = 1,\ldots, n \) . Furthermore, \( {NF}\left( w\right) = \) \( {NF}\left( {x}_{1}\r... | Proof. The existence of the unique factorization \( w = {x}_{1}\cdots {x}_{n} \) is a consequence of Corollary 2.4.6. It remains only to prove the statement about normal forms.\n\nIf \( w \in {W}_{\left\lbrack n - 1\right\rbrack } \), the result holds by induction, since then \( {x}_{n} = e \in {}^{\left\lbrack n - 1\r... | Yes |
Example 3.4.4 For instance, order the generators of the group \( {F}_{4} \) as shown in Figure 3.6. | The corresponding normal form forest is shown in Figure 3.7. We see from a glance at this forest, for example, that 13241213 is a normal form, whereas 21412141 is not. | No |
Lemma 3.4.5 Let \( w \in W \) be distinguished and \( s \in S \) . Then, we have the following:\n\n(i) If \( s \in {D}_{R}\left( w\right) \), then \( {ws} \) is distinguished.\n\n(ii) If \( s \notin {D}_{R}\left( w\right) \), then either ws is distinguished or there is \( {s}^{\prime } \in S \smallsetminus \left\{ {s}_... | Proof. We have that \( \ell \left( {ws}\right) = \ell \left( w\right) - 1 \) and \( \ell \left( {{s}^{\prime }w}\right) = \ell \left( w\right) + 1 \) if \( {s}^{\prime } \neq {s}_{n} \) . Hence, \( \ell \left( {{s}^{\prime }{ws}}\right) = \ell \left( w\right) > \ell \left( {ws}\right) \) for all \( {s}^{\prime } \neq {... | No |
Lemma 3.4.6 Let \( w \in W \) be distinguished, and \( s \in S \) (excluding the case \( w = s = {s}_{n} \) ). Then, there exists \( j \in \left\lbrack {n - 1}\right\rbrack \) such that \[ {NF}\left( {ws}\right) = \left\{ \begin{array}{ll} {nNF}\left( {{s}_{n}{ws}}\right) , & \text{ if }{ws} \in {}^{\left\lbrack n - 1\... | Proof. If \( {ws} \in {}^{\left\lbrack n - 1\right\rbrack }W \), then the result follows immediately from Lemma 3.4.1. If \( {ws} \notin {}^{\left\lbrack n - 1\rbrack }W \), then, by Lemma 3.4.5, there exists \( {s}^{\prime } \in S \smallsetminus \left\{ {s}_{n}\right\} \) such that \( {ws} = {s}^{\prime }w \) . Howeve... | Yes |
Theorem 3.4.7 Let \( v \in W \) and \( s \in S \). Then, there exists \( {s}^{\prime } \in S \smallsetminus \left\{ {s}_{n}\right\} \) such that \[ {NF}\left( {vs}\right) = \left\{ \begin{array}{ll} {NF}\left( u\right) , & \text{ if }w = s = {s}_{n}, \\ {NF}\left( u\right) {nNF}\left( {{s}_{n}{ws}}\right) , & \text{ if... | Proof. If \( {ws} \in {}^{\left\lbrack n - 1\right\rbrack }W \), then \( {\left( vs\right) }_{\left\lbrack n - 1\right\rbrack } = u \) and \( {}^{\left\lbrack n - 1\right\rbrack }\left( {vs}\right) = {ws} \), and the result follows from Proposition 3.4.2 and Lemma 3.4.6. If \( {ws} \notin {}^{\left\lbrack n - 1\right\r... | Yes |
Lemma 4.1.1 The coordinates \( \left( {q,{q}^{\prime }}\right) \) of a point \( {q\beta } + {q}^{\prime }{\beta }^{\prime } \) are transformed as follows by the orthogonal reflections: | Proof. We have that \( r\left( {1,0}\right) = \left( {1,0}\right) \) and \( {r}^{\prime }\left( {0,1}\right) = \left( {0,1}\right) \), since points on the reflecting lines remain fixed. The geometry indicated in Figure 4.1 shows that \( {r}^{\prime }\left( {1,0}\right) = \left( {-1,\frac{2\left| \beta \right| \cos \gam... | Yes |
Theorem 4.1.3 Let \( \left( {W, S}\right) \) be a Coxeter system. Then, the mapping \( s \mapsto \) \( {\sigma }_{s}^{ * }\left( {s \in S}\right) \) extends uniquely to a homomorphism \( {\sigma }^{ * } : W \rightarrow {GL}\left( {\mathbb{R}}^{S}\right) \) . | Proof of Proposition 1.1.1. The homomorphism \( {\sigma }^{ * } \) maps \( s \) and \( {s}^{\prime } \) to distinct elements in \( {GL}\left( {\mathbb{R}}^{S}\right) \), whose product has order \( m\left( {s,{s}^{\prime }}\right) \) . This proves (i) and that \( m\left( {s,{s}^{\prime }}\right) \), if finite, divides t... | No |
Proposition 4.2.1 For all \( s,{s}^{\prime } \in S \), the following hold:\n\n(i) \( {\sigma }_{s}^{2} = \mathrm{{id}} \) .\n\n(ii) The order of \( {\sigma }_{s}{\sigma }_{{s}^{\prime }} \) is \( m\left( {s,{s}^{\prime }}\right) \) . | Proof. Let \( {\sum }_{s} \) be the matrix representing \( {\sigma }_{s} \) in the \( \alpha \) basis; that is, for all \( \beta \in V \)\n\n\[{\sigma }_{s}\left( \beta \right) = {\sum }_{s}\beta\]\n\n(4.15)\n\nDefinitions (4.10) and (4.14) show that \( {\sum }_{s} = I + {K}_{s} \), where \( I \) is the identity matrix... | No |
Theorem 4.2.2 The mapping \( s \mapsto {\sigma }_{s} \) has a unique extension to a homomorphism \( \sigma : w \mapsto {\sigma }_{w} \) from \( W \) to \( {GL}\left( V\right) \) . | The linear mapping \( {\sigma }_{w} : V \rightarrow V \) is defined by \( {\sigma }_{w} = {\sigma }_{{s}_{1}}{\sigma }_{{s}_{2}}\ldots {\sigma }_{{s}_{k}} \) for any expression \( w = {s}_{1}{s}_{2}\ldots {s}_{k},{s}_{i} \in S \) . Its action on a vector \( \beta \in V \) will be notationally simplified to \[ w\left( \... | Yes |
Proposition 4.2.3 For all \( w \in W,\beta \in V \) and \( p \in {V}^{ * } \), \[ \langle w\left( p\right) \mid \beta \rangle = \left\langle {p \mid {w}^{-1}\left( \beta \right) }\right\rangle . \] | Proof. Equations (4.15) and (4.16) give that \[ \langle s\left( p\right) \mid \beta \rangle = {\left( {\sum }_{s}^{tr}p\right) }^{tr}\beta = {p}^{tr}{\sum }_{s}\beta = \langle p \mid s\left( \beta \right) \rangle . \] The formula for \( \ell \left( w\right) > 1 \) follows by repeated application. \( ▱ \) | No |
Proposition 4.2.5 For all \( w \in W \) and \( s \in S \), the following hold:\n\n(i) \( \ell \left( {ws}\right) > \ell \left( w\right) \) implies \( w\left( {\alpha }_{s}\right) > 0 \) .\n\n(ii) \( \ell \left( {ws}\right) < \ell \left( w\right) \) implies \( w\left( {\alpha }_{s}\right) < 0 \) . | Proof. We prove part (i) by induction on \( \ell \left( w\right) \), the \( \ell \left( w\right) = 0 \) case being clearly true.\n\nSo, assume that \( \ell \left( {ws}\right) > \ell \left( w\right) > 0 \), and let \( {s}^{\prime } \in {D}_{R}\left( w\right) \) . Put \( J = \left\{ {s,{s}^{\prime }}\right\} \) , and let... | Yes |
Corollary 4.2.6 Let \( p \in {\mathbb{R}}_{ + }^{S} \subseteq {V}^{ * } \), and \( u, v \in W \) . Then, the following hold:\n\n(i) \( {D}_{L}\left( u\right) = \left\{ {s \in S : \left\langle {u\left( p\right) \mid {\alpha }_{s}}\right\rangle }\right. \) is negative \( \} \) .\n\n(ii) If \( u \neq v \), then \( u\left(... | Proof. Since all coefficients of \( p \) are positive, Proposition 4.2.5 implies that \( \left\langle {u\left( p\right) \mid {\alpha }_{s}}\right\rangle = \left\langle {p \mid {u}^{-1}\left( {\alpha }_{s}\right) }\right\rangle \) is negative if and only if \( \ell \left( {{u}^{-1}s}\right) < \ell \left( {u}^{-1}\right)... | Yes |
Theorem 4.3.1 Consider play sequences starting from some positive position \( p \in {\mathbb{R}}_{ + }^{S} \) .\n\n(i) Two play sequences \( {s}_{1}{s}_{2}\ldots {s}_{k} \) and \( {s}_{1}^{\prime }{s}_{2}^{\prime }\ldots {s}_{q}^{\prime } \) lead to the same position (i.e., \( {p}^{{s}_{1}{s}_{2}\ldots {s}_{k}} = {p}^{... | Proof. The rule for changing the \ | No |
Example 4.4.2 It is instructive to visualize the \( W \) -action on \( V \) and its induced root system, and to compare it to the geometry of the \( W \) -action on \( {V}^{ * } \), in the accessible case of rank 2. | Figure 4.2 shows the case of the Weyl group \( {G}_{2} = {I}_{2}\left( 6\right) = { \circ }_{a}^{6}{ \circ }_{b} \), for which \( \left( {{\alpha }_{a} \mid {\alpha }_{b}}\right) = - \cos \frac{\pi }{6} \) . The picture of the space \( V \) indicates the six positive roots and how they are generated by simple reflectio... | Yes |
Lemma 4.4.3 The mapping \( s \) permutes the set \( {\Phi }^{ + } \smallsetminus \left\{ {\alpha }_{s}\right\} \) . | Proof. Let \( \gamma \in {\Phi }^{ + } \smallsetminus \left\{ {\alpha }_{s}\right\} \) . From statement (4.27) follows that \( \gamma \) is not a scalar multiple of \( {\alpha }_{s} \) . Thus, \( \left\langle {{\alpha }_{{s}^{\prime }}^{ * } \mid \gamma }\right\rangle \) is positive for some \( {s}^{\prime } \neq s \) ... | Yes |
Proposition 4.4.4 For all \( w \in W \) , \n\n\[ \n\ell \left( w\right) = \operatorname{card}N\left( w\right) \n\] | Proof. The formula holds for \( \ell \left( w\right) \leq 1 \) by Lemma 4.4.3. We continue by induction on \( \ell \left( w\right) \) . \n\nSuppose that \( w = {us} > u, s \in S \) . Equation (4.25) and Lemma 4.4.3 show for \( \beta \in {\Phi }^{ + } \) that \n\n\[ \nw\left( \beta \right) \in {\Phi }^{ - }\; \Leftright... | No |
Proposition 4.4.5 The mapping \( \rho : \gamma \mapsto {t}_{\gamma } \) is a bijective correspondence between positive roots and reflections. | It follows readily from our earlier computations that\n\n\[ \nw\left( \gamma \right) = \beta \; \Rightarrow \;w{t}_{\gamma }{w}^{-1} = {t}_{\beta }, \]\n\n(4.32)\n\nfor all \( w \in W \) and \( \beta ,\gamma \in {\Phi }^{ + } \) . Namely, letting \( \gamma = u\left( {\alpha }_{s}\right) \), we have\n\n\[ \nw{t}_{\gamma... | No |
Proposition 4.4.6 For all \( w \in W \) and \( \gamma \in {\Phi }^{ + } \) , | Proof. Suppose that \( \ell \left( {w{t}_{\gamma }}\right) < \ell \left( w\right) \), and let \( w = {s}_{1}{s}_{2}\ldots {s}_{k} \) be a reduced expression. Then, by Corollary 1.4.4, \( {t}_{\gamma } = {s}_{k}{s}_{k - 1}\ldots {s}_{i}\ldots {s}_{k - 1}{s}_{k} \) for some \( i \in \left\lbrack k\right\rbrack \) and, he... | Yes |
Lemma 4.5.1 Let \( p \in C \) . Then,\n\n\[ \operatorname{Stab}\left( p\right) = {W}_{J} \]\n\nwhere \( J\overset{\text{ def }}{ = }\left\{ {s \in S : \left\langle {p \mid {\alpha }_{s}}\right\rangle = 0}\right\} \) . | Proof. If \( s \in J \), then\n\n\[ \langle s\left( p\right) \mid \beta \rangle = \langle p \mid s\left( \beta \right) \rangle = \left\langle {p \mid \beta - 2\left( {{\alpha }_{s} \mid \beta }\right) {\alpha }_{s}}\right\rangle = \langle p \mid \beta \rangle \]\n\nfor all \( \beta \in V \) and, hence, \( s \in \operat... | Yes |
Lemma 4.5.2 Let \( p \in {V}^{ * } \) be such that \( M\left( p\right) \) is finite. Then, there exists a \( w \in W \) such that \( w\left( p\right) \in C \) . | Proof. Let \( p \in {V}^{ * } \smallsetminus C \) be such that \( 0 < \left| {M\left( p\right) }\right| < \infty \) . Then, there exists \( s \in S \) such that \( {\alpha }_{s} \in M\left( p\right) \) . However, it is clear from our definitions that\n\n\[ s\left( {M\left( {s\left( p\right) }\right) }\right) \subseteq ... | Yes |
Theorem 4.5.3 Let \( H \) be a finite subgroup of \( W \) . Then, there exists \( w \in \) \( W \) and \( J \subseteq S \) such that \( {W}_{J} \) is finite and \( {wH}{w}^{-1} \subseteq {W}_{J} \) . | Proof. We proceed by induction on \( \left| S\right| \), the result being clear if \( \left| S\right| = 1 \) . We may assume that \( \left| W\right| = \infty \), otherwise there is nothing to prove.\n\nLet\n\n\[ p\overset{\text{ def }}{ = }\mathop{\sum }\limits_{{s \in S}}{\alpha }_{s}^{ * }\;\text{ and }\;q\overset{\t... | Yes |
Proposition 4.5.4 Let \( \alpha ,\beta \in {\Phi }^{ + } \). (i) If \( \left| \left( {\alpha \mid \beta }\right) \right| < 1 \), then the subgroup generated by \( {t}_{\alpha } \) and \( {t}_{\beta } \) is a finite dihedral group. | Proof. Assume that \( \left| \left( {\alpha \mid \beta }\right) \right| < 1 \). Let \( w \in W \) be such that \( w\left( \beta \right) \in \Pi \). Since \( \left( {\alpha \mid \beta }\right) = \left( {w\left( \alpha \right) \mid w\left( \beta \right) }\right) ,{t}_{w\left( \alpha \right) } = w{t}_{\alpha }{w}^{-1},{t}... | Yes |
Proposition 4.5.5 The set\n\n\[\n\\left\\{ {\\left( {\\alpha \\mid {\\alpha }_{s}}\\right) : \\alpha \\in {\\Phi }^{ + }, s \\in S,\\left| \\left( {\\alpha \\mid {\\alpha }_{s}}\\right) \\right| < 1}\\right\\}\n\]\n\n(4.38)\n\nis finite. | Proof. Let\n\n\[\n{\\mathcal{P}}_{\\text{fin }}\\overset{\\text{ def }}{ = }\\left\\{ {J \\subseteq S : \\left| {W}_{J}\\right| < \\infty }\\right\\}\n\]\n\n\[\n{\\mathcal{A}}_{\\text{fin }}\\overset{\\text{ def }}{ = }\\left\\{ {\\left( {\\alpha \\mid \\beta }\\right) : \\alpha ,\\beta \\in \\Phi ,{t}_{\\alpha },{t}_{... | Yes |
Lemma 4.6.2 Let \( s \in S \) and \( \beta \in {\Phi }^{ + } - \left\{ {\alpha }_{s}\right\} \) . Then, \[ \operatorname{dp}\left( {s\left( \beta \right) }\right) = \left\{ \begin{array}{ll} \operatorname{dp}\left( \beta \right) - 1, & \text{ if }\left( {\beta \mid {\alpha }_{s}}\right) > 0, \\ \operatorname{dp}\left( ... | Proof. If \( \left( {\beta \mid {\alpha }_{s}}\right) = 0 \), then \( s\left( \beta \right) = \beta \), so trivially \( \operatorname{dp}\left( {s\left( \beta \right) }\right) = \operatorname{dp}\left( \beta \right) \) . Suppose that \( \left( {\beta \mid {\alpha }_{s}}\right) > 0 \) . Clearly, \( \operatorname{dp}\lef... | Yes |
Lemma 4.6.4 \( s\left( \beta \right) > \beta \Leftrightarrow {B}_{s} > {b}_{s} \) . | Proof. It is clear from our definitions that \( {B}_{s} - {b}_{s} = - 2\left( {{\alpha }_{s} \mid \beta }\right) \) . Now use Lemma 4.6.2. \( ▱ \) | No |
Lemma 4.7.1 Let \( \alpha \in \sum \) and \( s \in S,\alpha \neq {\alpha }_{s} \) . Then, \( \left( {\alpha \mid {\alpha }_{s}}\right) < 1 \) . | Proof. We proceed by induction on \( \mathrm{{dp}}\left( \alpha \right) \), the result being clear if \( \mathrm{{dp}}\left( \alpha \right) \) \( = 1 \) .\n\nSuppose \( \operatorname{dp}\left( \alpha \right) \geq 2 \) . Since \( \alpha \in \sum \), there is \( \beta \in \sum \) and \( r \in S \) such that \( \beta \var... | Yes |
Lemma 4.7.2 Let \( \beta ,\gamma \in \sum \) be such that \( \beta \vartriangleleft \gamma \) in the root poset, \( \operatorname{dp}\left( \beta \right) \geq 2 \) . Then, \( \mathcal{N}\left( \beta \right) \supseteq \mathcal{N}\left( \gamma \right) \) . | Proof. Let \( s \in S \smallsetminus \mathcal{N}\left( \beta \right) \) . Then, we have from Lemma 4.7.1 that \( \left( {\beta \mid {\alpha }_{s}}\right) \leq \) -1 . On the other hand, since \( \gamma = r\left( \beta \right) \) for some \( r \in S \) and \( \gamma > \beta \), we conclude from Lemma 4.7.1 (applied to \... | Yes |
Lemma 4.7.4 Let \( \beta \in {\Phi }^{ + } \) and \( s \in S \) . Then, \( \beta \) dominates \( {\alpha }_{s} \) if and only if \( \left( {\beta \mid {\alpha }_{s}}\right) \geq 1 \) . | Proof. We may clearly assume that \( \beta \neq {\alpha }_{s} \) and, hence, that \( s\left( \beta \right) \in {\Phi }^{ + } \) . Suppose that \( \beta \) dominates \( {\alpha }_{s} \) and that \( \left( {\beta \mid {\alpha }_{s}}\right) < 1 \) . Since \( \beta \) dominates \( {\alpha }_{s} \) and \( {t}_{\beta }\left(... | Yes |
Lemma 4.7.5 Let \( \beta ,\alpha \in {\Phi }^{ + } \) be such that \( \beta \vartriangleleft \alpha \) in the root poset. Then, we have that the following hold:\n\n(i) If \( \beta \vartriangleleft \alpha \) is long, then \( \alpha \) is not humble.\n\n(ii) If \( \beta \vartriangleleft \alpha \) is short, then \( \alpha... | Proof. Let \( s \in S \) be such that \( \alpha = s\left( \beta \right) \) .\n\nAssume that (i) holds. Then, \( \left( {\alpha \mid {\alpha }_{s}}\right) \geq 1 \) and, hence, \( \alpha \) dominates \( {\alpha }_{s} \) by Lemma 4.7.4. So \( \alpha \) is not humble.\n\nAssume now that (ii) holds. Then, \( 0 > \left( {\b... | Yes |
Theorem 4.7.6 Let \( \alpha \in {\Phi }^{ + } \) . Then, \( \alpha \in \sum \) if and only if \( \alpha \) is humble. | Proof. Assume that \( \alpha \in \sum \) . Then, by definition, there is a saturated chain in the root poset, consisting entirely of short edges, from some simple root to \( \alpha \) . However, simple roots are humble, so \( \alpha \) is humble by part (ii) of Lemma 4.7.5.\n\nConversely, suppose that \( \alpha \) is h... | Yes |
Corollary 4.7.7 \( \sum \) is an order ideal in the root poset. | Proof. Let \( \alpha \in \sum \) and \( \beta \in {\Phi }^{ + } \) be such that \( \beta \vartriangleleft \alpha \) . By Theorem 4.7.6, \( \alpha \) is humble, and, hence, by Lemma 4.7.5, \( \beta \vartriangleleft \alpha \) is short and \( \beta \) is humble. Hence, by Theorem 4.7.6, \( \beta \in \sum \) . \( ▱ \) | Yes |
Lemma 4.8.1 Let \( \alpha \in \sum, s \in S \) and \( w \in W \) be such that \( s\left( \alpha \right) \in {\Phi }^{ + } \smallsetminus \sum \) and \( s \notin {D}_{R}\left( w\right) \) . Then, \( {ws}\left( \alpha \right) > 0 \) . | Proof. Since \( {ws} > w \), we have that \( w\left( {\alpha }_{s}\right) > 0 \) and, hence, \( {ws}\left( {\alpha }_{s}\right) < 0 \) . Suppose that \( {ws}\left( \alpha \right) < 0 \) . Since \( \alpha \in \sum \) but \( s\left( \alpha \right) \notin \sum \), and \( \sum \) is an order ideal in the root poset, we hav... | Yes |
Proposition 4.8.2 Let \( w \in W \) and \( s \notin {D}_{R}\left( w\right) \) . Then, \[ {D}_{\sum }\left( {ws}\right) = \left\{ {\alpha }_{s}\right\} \cup \left( {\left\{ {s\left( \beta \right) : \beta \in {D}_{\sum }\left( w\right) }\right\} \cap \sum }\right) . \] | Proof. Let \( \alpha \in {D}_{\sum }\left( {ws}\right) \) . Then, \( {ws}\left( \alpha \right) < 0 \) and, hence, by Lemma 4.8.1, \( s\left( \alpha \right) \notin {\Phi }^{ + } \smallsetminus \sum \) . Therefore, either \( s\left( \alpha \right) \in {\Phi }^{ - } \), which implies \( \alpha = {\alpha }_{s} \), or \( s\... | Yes |
Theorem 4.8.3 The language of reduced expressions is regular. | Proof. We construct a finite state automaton for the language of reduced expressions as follows. Take\n\n\[ \mathcal{S}\overset{\text{ def }}{ = }\left\{ {{D}_{\sum }\left( w\right) : w \in W}\right\} \]\n\nas the set of nodes of the automaton, with \( {D}_{\sum }\left( e\right) \left( { = \varnothing }\right) \) as th... | Yes |
To illustrate the general construction of a finite state automaton in the proof, we end this section with an example. Namely, we construct the canonical automaton for the group \( \overset{ \circ }{a}\overset{ \circ }{b}\overset{ \circ }{c} \) . | The first step is to determine the set of small roots and the edge-labeled order ideal of the root poset that they determine. This task was achieved in Figure 4.7, from which we extract the simplified picture (Figure 4.10) of the edge-labeled order ideal.\n\n }\left( q\right) \) is rational. | Proof. We have that \( {r}_{k} \) equals the number of directed paths of length \( k \) in the automaton from the start node to any node in \( \mathcal{S} \) . Since \( \left| \mathcal{S}\right| < \infty \) the result follows via the well-known \ | No |
Theorem 4.9.4 Let \( \\left( {W, S}\\right) \) be an affine Weyl group, with corresponding finite Weyl group \( \\bar{W} \) . Let \( t \) be the number of reflections of \( \\bar{W} \) . Then, we have the following:\n\n(i) \( \\left| \\sum \\right| = {2t} \) .\n\n(ii) The number of connected regions of \( U \\smallsetm... | The second part of the theorem is due to Shi [456], who introduced and studied arrangements of small hyperplanes in the affine case. In fact, in that case, these arrangements are usually known as Shi arrangements. | No |
Theorem 5.1.1 There is a unique family of polynomials \( {\left\{ {R}_{u, v}\left( q\right) \right\} }_{u, v \in W} \subseteq \) \( \mathbb{Z}\left\lbrack q\right\rbrack \) satisfying the following conditions:\n\n(i) \( {R}_{u, v}\left( q\right) = 0,\; \) if \( u \nleq v \) .\n\n(ii) \( {R}_{u, v}\left( q\right) = 1,\;... | The uniqueness part of this theorem is trivial. What is not obvious is the existence. This follows from the invertibility of certain basis elements of the Hecke algebra \( \mathcal{H} \) of \( W \) and is proved in \( §§{7.4} \) and 7.5 of [306]. | Yes |
Suppose that we want to compute \( {R}_{{123},{321}}\left( q\right) \) in \( {S}_{3} \). | Choosing \( s = \left( {1,2}\right) \in {D}_{R}\left( {321}\right) \), we have from part (iii) of Theorem 5.1.1 that\n\n\[ \n{R}_{{123},{321}}\left( q\right) = q{R}_{{213},{231}}\left( q\right) + \left( {q - 1}\right) {R}_{{123},{231}}\left( q\right) .\n\]\n\nNow, choosing \( s = \left( {2,3}\right) \in {D}_{R}\left( {... | Yes |
Proposition 5.1.3 Let \( u, v \in W, u \leq v \) . Then, \( {R}_{u, v}\left( q\right) \) is a monic polynomial of degree \( \ell \left( {u, v}\right) \) and with constant term \( {\left( -1\right) }^{\ell \left( {u, v}\right) } \) . | Proof. All three statements are true if \( \ell \left( v\right) = 0 \) (i.e., if \( v = e \) ) by part (ii) of Theorem 5.1.1. So let \( \ell \left( v\right) > 0 \) and assume by induction that they hold whenever the second indexing element has length \( < \ell \left( v\right) \) . Let \( s \in {D}_{R}\left( v\right) \)... | Yes |
Theorem 5.1.4 There is a unique family of polynomials \( {\left\{ {P}_{u, v}\left( q\right) \right\} }_{u, v \in W} \subseteq \) \( \mathbb{Z}\left\lbrack q\right\rbrack \) satisfying the following conditions:\n\n(i) \( {P}_{u, v}\left( q\right) = 0,\; \) if \( u \nleq v \) .\n\n(ii) \( {P}_{u, v}\left( q\right) = 1,\;... | A proof of Theorem 5.1.4 appears in \( §§{7.9},{7.10} \), and 7.11 of [306]. | No |
Proposition 5.1.5 Let \( u, v \in W, u \leq v \) . Then, \( {P}_{u, v}\left( 0\right) = 1 \) . | Proof. We proceed by induction on \( \ell \left( {u, v}\right) \), the result being true by part (ii) of Theorem 5.1.4 if \( \ell \left( {u, v}\right) = 0 \) . So let \( \ell \left( {u, v}\right) > 0 \) . Then, we conclude from parts (iii) and (iv) of Theorem 5.1.4 that\n\n\[ 0 = \mathop{\sum }\limits_{{a \in \left\lbr... | Yes |
Let us compute \( {P}_{{123},{321}}\left( q\right) \) in \( {S}_{3} \) . | From part (iv), we deduce that\n\n\[ \n{q}^{3}{P}_{{123},{321}}\left( {q}^{-1}\right) - {P}_{{123},{321}}\left( q\right) = {R}_{{123},{213}}\left( q\right) {P}_{{213},{321}}\left( q\right) \n\] \n\n\[ \n+ {R}_{{123},{132}}\left( q\right) {P}_{{132},{321}}\left( q\right) \n\] \n\n\[ \n+ {R}_{{123},{231}}\left( q\right) ... | Yes |
Theorem 5.1.7 Let \( u, v \in W, u \leq v \), and \( s \in {D}_{R}\left( v\right) \) . Then,\n\n\[ \n{P}_{u, v}\left( q\right) = {q}^{1 - c}{P}_{{us},{vs}}\left( q\right) + {q}^{c}{P}_{u,{vs}}\left( q\right) - \mathop{\sum }\limits_{\left\{ z : s \in {D}_{R}\left( z\right) \right\} }{q}^{\frac{\ell \left( {z, v}\right)... | A proof of this result can be found in [306, §7.11]. | No |
Proposition 5.1.8 Let \( u, v \in W, u \leq v \) . If \( s \in {D}_{R}\left( v\right) \), then\n\n\[ \n{P}_{u, v}\left( q\right) = {P}_{{us}, v}\left( q\right) \n\] | Proof. Applying Theorem 5.1.7 first to the pair \( \left( {u, v}\right) \), and then to \( \left( {{us}, v}\right) \) , one sees that the right-hand side of equation (5.4) does not change, except that \( {P}_{u, z}\left( q\right) \) gets replaced by \( {P}_{{us}, z}\left( q\right) \) . However, these two polynomials ar... | Yes |
Proposition 5.1.9 Let \( z, w \in W, z \leq w \), be such that \( \bar{\mu }\left( {z, w}\right) \neq 0 \) and \( \ell \left( {z, w}\right) > 1 \) . Then, \( {D}_{R}\left( z\right) \supseteq {D}_{R}\left( w\right) \) . | Proof. Let \( s \in {D}_{R}\left( w\right) \) and suppose that \( s \notin {D}_{R}\left( z\right) \) . Then, from Proposition 5.1.8 we conclude that \( {P}_{z, w}\left( q\right) = {P}_{{zs}, w}\left( q\right) \) . Hence,\n\n\[ \bar{\mu }\left( {z, w}\right) = \left\lbrack {q}^{\frac{1}{2}\left( {\ell \left( {z, w}\righ... | Yes |
Proposition 5.2.1 There exists a reflection ordering on \( {\Phi }^{ + } \) . | Proof. Fix an indexing (i.e., a total ordering) of the elements of \( S \), say \( S = \left\{ {{s}_{1},\ldots ,{s}_{n}}\right\} \) . Let\n\n\[ \mathcal{U}\overset{\text{ def }}{ = }\left\{ {\mathop{\sum }\limits_{{i = 1}}^{n}{c}_{{s}_{i}}{\alpha }_{{s}_{i}} : \mathop{\sum }\limits_{{i = 1}}^{n}{c}_{{s}_{i}} = 1}\right... | Yes |
Proposition 5.2.2 Let \( < \) be a reflection ordering, \( s \in S \), and \( \beta \in {\Phi }^{ + } \smallsetminus \) \( \left\{ {\alpha }_{s}\right\} \) . Then, \( \beta < {\alpha }_{s} \) if and only if \( s\left( \beta \right) < {\alpha }_{s} \) . | Proof. We may assume that \( s\left( \beta \right) \neq \beta \) and, hence, that \( \left( {\beta \mid {\alpha }_{s}}\right) \neq 0 \) . If \( \left( {{\alpha }_{s} \mid \beta }\right) < 0 \), then since \( s\left( \beta \right) = \beta - 2\left( {{\alpha }_{s} \mid \beta }\right) {\alpha }_{s} \), there follows from ... | Yes |
Proposition 5.2.3 Let \( < \) be a reflection ordering, and \( s \in S \) . Then, \( { < }_{s} \) and \( { < }^{s} \) are also reflection orderings. | Proof. We show that \( { < }^{s} \) is a reflection ordering, the proof for \( { < }_{s} \) being entirely similar.\n\nLet \( \alpha ,\beta \in {\Phi }^{ + } \) and \( a, b \in {\mathbb{R}}_{ > 0} \) be such that \( {a\alpha } + {b\beta } \in {\Phi }^{ + } \) . Suppose that \( \alpha < \beta \) . Then,\n\n\[ \alpha < {... | Yes |
Proposition 5.3.1 Let \( u, v \in W \) . Then, there exists a unique polynomial \( {\widetilde{R}}_{u, v}\left( q\right) \in \mathbb{N}\left\lbrack q\right\rbrack \) such that\n\n\[ \n{R}_{u, v}\left( q\right) = {q}^{\frac{\ell \left( {u, v}\right) }{2}}{\widetilde{R}}_{u, v}\left( {{q}^{\frac{1}{2}} - {q}^{-\frac{1}{2... | Proof. The result is trivially true if \( u \nleq v \), so we may assume that \( u \leq v \) . If \( f, g \in \mathbb{R}\left\lbrack q\right\rbrack \) are such that\n\n\[ \nf\left( {{q}^{\frac{1}{2}} - {q}^{-\frac{1}{2}}}\right) = g\left( {{q}^{\frac{1}{2}} - {q}^{-\frac{1}{2}}}\right) \n\]\n\nfor all \( q \in {\mathbb... | Yes |
Proposition 5.3.2 Let \( u, v \in W, u \leq v \) . Then, \( {\widetilde{R}}_{u, v}\left( q\right) \) is a monic polynomial of degree \( \ell \left( {u, v}\right) \) . Furthermore, if \( s \in {D}_{R}\left( v\right) \), then | \[ {\widetilde{R}}_{u, v}\left( q\right) = \left\{ \begin{array}{ll} {\widetilde{R}}_{{us},{vs}}\left( q\right) , & \text{ if }s \in {D}_{R}\left( u\right) , \\ {\widetilde{R}}_{{us},{vs}}\left( q\right) + q{\widetilde{R}}_{u,{vs}}\left( q\right) , & \text{ if }s \notin {D}_{R}\left( u\right) .▱ \end{array}\right. \] | No |
Lemma 5.3.3 Let \( u, v \in W \) be such that \( u \rightarrow v \), and \( s \in S \smallsetminus \left\{ {{u}^{-1}v}\right\} \) . Then, \( {us} \rightarrow {vs} \) . | Proof. Since \( {vs} = {us}\left( {sts}\right) \) (if \( v = {ut}, t \in T \) ), it suffices to show that \( \ell \left( {us}\right) < \ell \left( {vs}\right) \) . This is clear if \( \ell \left( {u, v}\right) \geq 3 \) . If \( \ell \left( {u, v}\right) = 1 \), it follows from Proposition 2.2.7. \( ▱ \) | No |
Consider the two permutations \( u = {1234} \) and 4312 of \( {S}_{4} \) , and choose the reflection ordering\n\n\[ \left( {1,2}\right) < \left( {1,3}\right) < \left( {1,4}\right) < \left( {2,3}\right) < \left( {2,4}\right) < \left( {3,4}\right) . \]\n\nThen, \( B\left( {u, v}\right) \) is the labeled directed graph de... | Note that it is a consequence of Theorem 5.3.4 that the polynomial \( {\widetilde{R}}_{u, v}\left( q\right) \) contains only odd powers of \( q \) or only even powers of \( q \), depending on the parity of \( \ell \left( {u, v}\right) \) . | Yes |
Let \( W = {S}_{4}, u = {1234}, v = {4321} \) and take \( \left( {1,2,1,3,2,1}\right) \) as a reduced decomposition for \( v \). Then, | \[ \mathcal{D}{\left( \left( 1,2,1,3,2,1\right) \right) }_{u} = \{ \left( {-,-,-,-,-, - }\right) ,\left( {1,-,1,-,-, - }\right) , \left( {-,2,-,-,2, - }\right) ,\left( {-,-,1,-,-,1}\right) , \left( {1,2,-,-,2,1}\right) \} \] and, hence, \[ {\widetilde{R}}_{{1234},{4321}}\left( q\right) = {q}^{6} + 3{q}^{4} + {q}^{2}. \... | Yes |
Proposition 5.3.9 Let \( \left( {{s}_{1},\ldots ,{s}_{r}}\right) \in {S}^{r} \) be a reduced decomposition and \( \left( {{a}_{1},\ldots ,{a}_{r}}\right) \) be a subexpression of \( \left( {{s}_{1},\ldots ,{s}_{r}}\right) \). Then, the following are equivalent:\n\n(i) \( \left( {{a}_{1},\ldots ,{a}_{r}}\right) \in \mat... | Proof. Let \( \left( {{a}_{1},\ldots ,{a}_{r}}\right) \in \mathcal{D}\left( {{s}_{1},\ldots ,{s}_{r}}\right) \) and \( 2 \leq j \leq r \). If \( {a}_{j} = {s}_{j} \), then (ii) clearly holds, so assume that \( {a}_{j} = e \). Then, since \( \left( {{a}_{1},\ldots ,{a}_{r}}\right) \in \) \( \mathcal{D}\left( {{s}_{1},\l... | Yes |
Proposition 5.4.1 Let \( \alpha \in C \) . Then,\n\n\[ \n{\Psi }_{\alpha }\left( q\right) = {\left( q - 1\right) }^{{\alpha }_{1}}{L}_{\frac{\left| \bar{\alpha }\right| - 1}{2}}\left( {{\Psi }_{\bar{\alpha }}\left( q\right) }\right) \n\]\n\n(5.26)\n\nif \( \ell \left( \alpha \right) \geq 2 \), and\n\n\[ \n{\Psi }_{\alp... | Proof. If \( \ell \left( \alpha \right) = 1 \), then \( T\left( \alpha \right) = \varnothing \) and equation (5.27) follows immediately from equation (5.24). So assume \( \ell \left( \alpha \right) \geq 2 \) and let \( \Gamma \in H\left( {T\left( \alpha \right) ,\left| \alpha \right| }\right) \) . Then,\n\n\( {\Gamma }... | Yes |
In \( W = {S}_{4} \), we have that\n\n\[ \n{R}_{{2134},{2431},{4321}}\left( q\right) = {R}_{{2134},{2431}}\left( q\right) {L}_{\frac{2 - 1}{2}}\left( {{R}_{{2431},{4321}}\left( q\right) }\right) \n\] | \[ \n= \left( {{q}^{3} - 2{q}^{2} + {2q} - 1}\right) {L}_{\frac{1}{2}}\left( {\left( q - 1\right) }^{2}\right) \n\]\n\n\[ \n= {q}^{3} - 2{q}^{2} + {2q} - 1 \n\]\n\nand, hence,\n\n\[ \n\left. {{R}_{{1234},{2134},{2431},{4321}}\left( q\right) }\right. = \left. {{R}_{{1234},{2134}}\left( q\right) {L}_{\frac{5 - 1}{2}}\lef... | Yes |
Theorem 5.5.2 Let \( u, v \in W, u < v \) . Then,\n\n\[ \n{P}_{u, v}\left( q\right) - {q}^{\ell \left( {u, v}\right) }{P}_{u, v}\left( \frac{1}{q}\right) = \mathop{\sum }\limits_{{\mathcal{C} \in C\left( {u, v}\right) }}{\left( -1\right) }^{\ell \left( \mathcal{C}\right) }{R}_{\mathcal{C}}\left( q\right) ,\n\]\n\nwhere... | Proof. We prove the result by induction on \( \ell \left( {u, v}\right) \) . If \( \ell \left( {u, v}\right) = 1 \), then\n\n\[ \n\mathop{\sum }\limits_{{\mathcal{C} \in C\left( {u, v}\right) }}{\left( -1\right) }^{\ell \left( \mathcal{C}\right) }{R}_{\mathcal{C}}\left( q\right) = - {R}_{u, v}\left( q\right) = 1 - q = ... | Yes |
Corollary 5.5.3 Let \( u, v \in W, u < v \) . Then,\n\n\[ \n{P}_{u, v}\left( q\right) = {L}_{\frac{\ell \left( {u, v}\right) - 1}{2}}\left( {\mathop{\sum }\limits_{{\mathcal{C} \in C\left( {u, v}\right) }}{\left( -1\right) }^{\ell \left( \mathcal{C}\right) }{R}_{\mathcal{C}}\left( q\right) }\right) .\n\] | \[ ▱ \] | No |
Proposition 5.5.4 Let \( u, v \in W, u \leq v \), and \( {\alpha }_{1},\ldots ,{\alpha }_{r} \in \mathbb{P}, r \geq 2 \) . Then,\n\n\[ \n{c}_{{\alpha }_{1},\ldots ,{\alpha }_{r}}\left( {u, v}\right) = \mathop{\sum }\limits_{{u \leq a \leq v}}{c}_{{\alpha }_{1},\ldots ,{\alpha }_{r - 1}}\left( {u, a}\right) {c}_{{\alpha... | Proof. Let \( \Delta \overset{\text{ def }}{ = }\left( {{a}_{0},\ldots ,{a}_{{\beta }_{r}}}\right) \in B\left( {u, v}\right) \) be such that \( D\left( {\Delta , < }\right) \subseteq \left\{ {{\beta }_{1},\ldots }\right. \) , \( \left. {\beta }_{r - 1}\right\} \), where \( {\beta }_{i}\overset{\text{ def }}{ = }{\alpha... | Yes |
Proposition 5.5.5 Let \( u, v \in W, u \leq v \), and \( \alpha \in C \) . Then,\n\n\[ \n{c}_{\alpha }\left( {u, v}\right) = \mathop{\sum }\limits_{{\left( {{a}_{0},\ldots ,{a}_{r}}\right) \in {C}_{r}\left( {u, v}\right) }}\mathop{\prod }\limits_{{j = 1}}^{r}\left\lbrack {q}^{{\alpha }_{j}}\right\rbrack \left( {\wideti... | Proof. We proceed by induction on \( r \in \mathbb{P} \) . If \( r = 1 \) then, by equation (5.33) and Theorem 5.3.4, we have that\n\n\[ \n{c}_{{\alpha }_{1}}\left( {u, v}\right) = \left\lbrack {q}^{{\alpha }_{1}}\right\rbrack \left( {\widetilde{R}}_{u, v}\right) \n\]\n\n(5.34)\n\n\nas desired. If \( r \geq 2 \), then ... | Yes |
Proposition 5.5.6 Let \( {a}_{0} < {a}_{1} < \cdots < {a}_{i} \) be a chain in \( W \) . Then,\n\n\[ \n{R}_{{a}_{0},\ldots ,{a}_{i}}\left( q\right) = \mathop{\sum }\limits_{{\alpha \in {\mathbb{P}}^{i}}}{q}^{\frac{\ell \left( {{a}_{0},{a}_{i}}\right) - \left| \alpha \right| }{2}}{\Psi }_{\alpha }\left( q\right) \mathop... | Proof. If \( i = 1 \), then equation (5.35) follows from equations (5.27),(5.29) and (5.10). We now proceed by induction on \( i \in \mathbb{P} \) . Since \( i \geq 2 \), we have from definition (5.28) and our induction hypothesis that\n\n\[ \n{R}_{{a}_{0},\ldots ,{a}_{i}}\left( q\right)\n\]\n\n\[ \n= {R}_{{a}_{0},{a}_... | Yes |
Theorem 5.5.7 Let \( < \) be a reflection ordering, \( u, v \in W, u < v \) . Then,\n\n\[ \n{P}_{u, v}\left( q\right) - {q}^{\ell \left( {u, v}\right) }{P}_{u, v}\left( \frac{1}{q}\right) = \mathop{\sum }\limits_{{\Delta \in B\left( {u, v}\right) }}{q}^{\frac{\ell \left( {u, v}\right) - \ell \left( \Delta \right) }{2}}... | Proof. From Theorem 5.5.2 and Propositions 5.5.5 and 5.5.6 we have that\n\n\[ \n{P}_{u, v}\left( q\right) - {q}^{\ell \left( {u, v}\right) }{P}_{u, v}\left( \frac{1}{q}\right) = \mathop{\sum }\limits_{{\mathcal{C} \in C\left( {u, v}\right) }}{\left( -1\right) }^{\ell \left( \mathcal{C}\right) }{R}_{\mathcal{C}}\left( q... | Yes |
Example 5.5.8 Let \( u = {2147563} \) and \( v = {6157243} \) . Considering all of the directed paths from 2147563 to 6157243 in Figure 5.1 (there are 62 of them) and computing the descent composition of each one, we obtain that \[ {P}_{{2147563},{6157243}}\left( q\right) - {q}^{5}{P}_{{2147563},{6157243}}\left( {q}^{-... | \[ = q\left( {2{\Upsilon }_{3} + 3{\Upsilon }_{2,1} + 3{\Upsilon }_{1,2} + 2{\Upsilon }_{1,1,1}}\right) \] \[ + {q}^{0}\left( {{\Upsilon }_{5} + 2{\Upsilon }_{4,1} + 4{\Upsilon }_{3,2} + 4{\Upsilon }_{2,3} + 2{\Upsilon }_{1,4} + 3{\Upsilon }_{3,1,1}} \right. \] \[ + 4{\Upsilon }_{1,3,1} + 3{\Upsilon }_{1,1,3} + 6{\Upsi... | Yes |
Proposition 5.6.1 Let \( \left( {W, S}\right) \) be a Coxeter system, \( u, v \in W, u \leq v \), and \( s \in D\left( v\right) \smallsetminus D\left( u\right) \) . Let\n\n\[ M\left( x\right) \overset{\text{ def }}{ = }{xs} \]\n\nfor all \( x \in \left\lbrack {u, v}\right\rbrack \) . Then, \( M \) is a special matching... | Proof. This follows immediately from the definition of a special matching and the Lifting Property. \( ▱ \) | No |
Corollary 5.6.2 Every lower Bruhat interval \( \left\lbrack {e, v}\right\rbrack \) has a special matching. | There is, of course, a left version of Proposition 5.6.1. Note that the converse is not true. Namely there are special matchings of Bruhat intervals which are not given by right or left multiplication by a simple reflection. For example, let \( \left( {W, S}\right) \) be a Coxeter system such that \( \left| S\right| \g... | No |
Let \( P = \left\lbrack {e, v}\right\rbrack \) be the lower Bruhat interval whose Hasse diagram is depicted in Figure 5.4. The elements of \( P \) are labeled with the integers from 1 to 18 . In order to use Theorem 5.6.3, we need to find a special matching \( M \) of \( P \) . | Suppose \( M\left( 1\right) = 2 \) . We have two possible choices for \( M\left( 3\right) \), namely 7 and 8, and two for \( M\left( 4\right) \), namely 5 and 6 . Suppose we choose \( M\left( 3\right) = 7 \) and \( M\left( 4\right) = 5 \) . These choices force\n\n\[ M = \{ \{ 1,2\} ,\{ 3,7\} ,\{ 4,5\} ,\{ 6,{11}\} ,\{ ... | Yes |
Lemma 6.2.2 Let \( x, y \in W, x < y \) . Then, the following hold:\n\n(i) If \( \{ x, y\} \in E \), then \( \ell \left( {x, y}\right) = \ell \left( y\right) - \ell \left( x\right) \) is odd.\n\n(ii) If \( \ell \left( {x, y}\right) = 1 \), then \( \{ x, y\} \in E \) and \( \bar{\mu }\left( {x, y}\right) = 1 \) .\n\n(ii... | Proof. The first two statements follow directly from the definitions. For the third, note that \( {P}_{x, y} = {P}_{{sx}, y} \) by implication (6.5), so the degree of \( {P}_{x, y} \) is too small. [Remark: This is a restatement of Proposition 5.1.9.] \( ▱ \) | No |
Lemma 6.2.4 Suppose that \( \ell \left( x\right) > \ell \left( y\right) \) . Then, \( x \rightarrow y \) is an edge in \( {\widetilde{\Gamma }}_{\left( W, S\right) } \) if and only if \( x{ \vartriangleright }_{L}y \) . | Proof. If \( x = {sy} \), then clearly \( x \rightarrow y \) is an \( s \) -labeled edge in \( {\widetilde{\Gamma }}_{\left( W, S\right) } \) . The converse follows from Lemma 6.2.2(iii). | Yes |
Proposition 6.2.7 If \( x{ \preccurlyeq }_{L}y \), then \( {D}_{R}\left( x\right) \supseteq {D}_{R}\left( y\right) \) . Consequently, if \( x{ \sim }_{L}y \), then \( {D}_{R}\left( x\right) = {D}_{R}\left( y\right) \) . | Proof. We may assume that \( x\xrightarrow[s]{\bar{\mu }}y \) is an edge in \( {\widetilde{\Gamma }}_{\left( W, S\right) } \) . There are two cases to consider:\n\n(i) \( y < x \) . Then, \( x = {sy} \), by Lemma 6.2.4, and hence \( {D}_{R}\left( y\right) \subseteq {D}_{R}\left( x\right) \) .\n\n(ii) \( y > x \) . Assu... | No |
Proposition 6.2.8 If \( x\overset{\bar{\mu }}{ \rightarrow }y \) is an edge in \( {\Gamma }_{\left( W, S\right) } \), then also \( {w}_{0}x\overset{\bar{\mu }}{ \rightarrow }{w}_{0}y \) , \( x{w}_{0}\overset{\bar{\mu }}{ = }y{w}_{0} \), and \( {w}_{0}x{w}_{0}\overset{\bar{\mu }}{ = }{w}_{0}y{w}_{0} \) are edges (all wi... | Proof. This follows from Corollary 2.3.3 and Exercise 5.13. \( ▱ \) | No |
Proposition 6.2.9 If \( x\xrightarrow[s]{\bar{\mu }}y, x \neq y \), is a directed edge in \( {\widetilde{\Gamma }}_{\left( W, S\right) } \), then so also are the following:\n\n(i) \( {w}_{0}y\underset{{w}_{0}s{w}_{0}}{\overset{\mu }{ \rightarrow }}{w}_{0}x \) .\n\n(ii) \( y{w}_{0}\xrightarrow[s]{\bar{\mu }}x{w}_{0} \) ... | Proof. Immediate from Exercise 2.10 and the preceding proposition. | No |
Lemma 6.3.1 If \( w = {s}_{1}\ldots {s}_{k} \) (with \( {s}_{i} \in S \) ), then\n\n\[ \n{a}_{x, y}\left( w\right) = \sum \bar{\mu }\left( {{x}_{0},{x}_{1}}\right) \bar{\mu }\left( {{x}_{1},{x}_{2}}\right) \ldots \bar{\mu }\left( {{x}_{k - 1},{x}_{k}}\right) ,\n\]\n\nwhere the sum is over all paths \( x = {x}_{0}\under... | Proof. Equation (6.6) can be restated as\n\n\[ \n{a}_{x, y}\left( s\right) = \left\{ \begin{array}{ll} \bar{\mu }\left( {x, y}\right) , & \text{ if }\left( {x \rightarrow y}\right) \in A, \\ 0, & \text{ otherwise. } \end{array}\right.\n\]\n\n(6.7)\n\nNow use the fact that \( A\left( w\right) = A\left( {s}_{1}\right) \c... | No |
Lemma 6.3.2 Let \( {\mathcal{C}}^{1},\ldots ,{\mathcal{C}}^{k} \) be the left cells of \( W \), labeled so that if \( {\mathcal{C}}^{i}{ \preccurlyeq }_{L} \) \( {\mathcal{C}}^{j} \), then \( i < j \) . Arrange the columns and rows of \( A\left( w\right) \) cellwise in this order. Then, \( A\left( w\right) \) has upper... | Proof. This is clear from Lemma 6.3.1, the definition of left cells, and the definition of \( { \preccurlyeq }_{L} \) for left cells. | No |
Proposition 6.3.4 Let \( \\left( {W, S}\\right) \) be a finite Coxeter system. Then,\n\n\[ \n{\\operatorname{Reg}}_{W} \\cong {\\bigoplus }_{\\mathcal{C}}K{L}_{\\mathcal{C}} \n\]\n\nwhere \( \\mathcal{C} \) runs over all the left cells of \( W \) . | Proof. This is an immediate consequence of Lemma 6.3.2 and Maschke's theorem (see, e.g., [358, p. 21]). | No |
Proposition 6.3.5 Let \( \mathcal{C} \) be a left cell. Then, the following hold:\n\n(i) \( K{L}_{\mathcal{C}{w}_{0}} \cong {\varepsilon K}{L}_{\mathcal{C}} \) . | Proof. We begin with part (i). The key fact is that \( {\widetilde{\Gamma }}_{\mathcal{C}{w}_{0}} \) is obtained from \( {\widetilde{\Gamma }}_{\mathcal{C}} \) by reversing the direction of all arrows, keeping their color \( s \) and weight \( \bar{\mu } \) , except that the \( \pm 1 \) weights on loops are switched (c... | Yes |
Lemma 6.3.6 Let \( y \in {\mathcal{D}}_{J}^{S} \) and \( x \leq y \) . Then, the following hold:\n\n(i) \( a \leq y \), for all \( a \in x{W}_{J} \) . In particular, \( \left\lbrack {e, y}\right\rbrack \) is a union of left cosets of \( {W}_{J} \) .\n\n(ii) \( {P}_{a, y}\left( q\right) = {P}_{x, y}\left( q\right) \), f... | Proof. Use that the upper projection \( {\bar{P}}^{J} : W \rightarrow {\mathcal{D}}_{J}^{S} \) is order-preserving (Exercise 2.16). Then, \( a \leq {\bar{P}}^{J}\left( a\right) = {\bar{P}}^{J}\left( x\right) \leq {\bar{P}}^{J}\left( y\right) = y \), for all \( a \in x{W}_{J} \) .\n\nFor part (ii), let \( z = {\bar{P}}^... | No |
Lemma 6.3.7 (i) If \( y \in {\mathcal{D}}_{J}^{S} \), then \( {C}_{y} \in {\overline{\mathcal{H}}}_{J} \) . | Proof. Since \( q = 1 \), we have from equation (6.1) that \( {C}_{y} = \mathop{\sum }\limits_{{x \in \left\lbrack {e, y}\right\rbrack }}{\varepsilon }_{x}{\varepsilon }_{y}{P}_{x, y}\left( 1\right) {T}_{x} \) . Now, \( \left\lbrack {e, y}\right\rbrack \) is a (disjoint) union of left cosets of \( {W}_{J} \), say \( {A... | Yes |
Theorem 6.3.8 Let \( J \subseteq S \) . Then,\n\n\[ \n{\operatorname{Ind}}_{J}^{S}\left\lbrack 1\right\rbrack \cong {\bigoplus }_{{D}_{R}\left( \mathcal{C}\right) \supseteq J}{\varepsilon K}{L}_{\mathcal{C}} \cong {\bigoplus }_{{D}_{R}\left( \mathcal{C}\right) \subseteq S \smallsetminus J}K{L}_{\mathcal{C}}\n\] | Proof. The map \( \mathcal{C} \mapsto {w}_{0}\mathcal{C} \) is a bijection between those left cells whose right descent set contains \( J \) and those whose right descent set is contained in \( S \smallsetminus J \) . Hence, the second equivalence is directly implied by Proposition 6.3.5(ii).\n\nWe will prove the first... | No |
Lemma 6.4.1 Let \( x, y \in {S}_{n},\ell \left( x\right) < \ell \left( y\right) \), and \( 1 < i < n \) . Then, the following are equivalent:\n\n(i) \( x\underset{K}{\overset{i}{ \approx }}y \) .\n\n(ii) \( {xs} < x < x{s}^{\prime } = y < {ys} \), with \( \left\{ {s,{s}^{\prime }}\right\} = \left\{ {{s}_{i - 1},{s}_{i}... | Proof. We leave the verification to the reader. Lemma 6.2.4 is of use for showing (iii) \( \Rightarrow \) (ii). | No |
Lemma 6.4.4 Let \( x, y \in {S}_{n} \) and suppose that \( x\underset{K}{\overset{i}{ \approx }}y \) . Then, the following hold:\n\n(i) \( Q\left( x\right) \) and \( Q\left( y\right) \) differ by a transposition \( {s}_{i - 1} \) or \( {s}_{i} \) .\n\n(ii) \( Q\left( x\right) \) uniquely determines \( Q\left( y\right) ... | Proof. Since \( x\underset{K}{\overset{i}{ \approx }}y \) and using that \( x \) and \( Q\left( x\right) \) have the same descent sets, we see that \( \left| {D\left( {Q\left( x\right) }\right) \cap \{ i - 1, i\} }\right| = 1 \), and similarly for \( y \) . This shows that (i) implies (ii).\n\nTo prove (i), we have to ... | No |
Corollary 6.4.5 Let \( x, y \in {S}_{n} \) . If there exists a labeled Knuth path\n\n\[ x\underset{K}{\overset{{i}_{1}}{ \approx }}{x}_{1}\underset{K}{\overset{{i}_{2}}{ \approx }}{x}_{2}\underset{K}{\overset{{i}_{3}}{ \approx }}\cdots \underset{K}{\overset{{i}_{k}}{ \approx }}{x}_{k} = y \]\n\nthen \( Q\left( x\right)... | \( ▱ \) | No |
Lemma 6.4.7 (Edge Transport) Suppose \( x, y \in {\mathcal{{DES}}}_{R}\left( i\right) \) and \( \{ x, y\} \) is an edge in \( {\Gamma }_{{S}_{n}} \) . Then, \( \left\{ {{x}^{ * },{y}^{ * }}\right\} \) is also an edge in \( {\Gamma }_{{S}_{n}} \) and \( \bar{\mu }\left( {{x}^{ * },{y}^{ * }}\right) = \) \( \bar{\mu }\le... | Proof. We omit the proof, which amounts to somewhat tedious case-by-case checking. See Exercise 5 or [322, pp. 175-176]. \( ▱ \) | No |
Corollary 6.4.8 If \( x \in {\mathcal{{DES}}}_{R}\left( i\right) \) and \( x\underset{L}{ \sim }y \), then \( y \in {\mathcal{{DES}}}_{R}\left( i\right) \) and \( {x}^{ * }\underset{L}{ \sim }{y}^{ * } \) | Proof. Since \( x\underset{L}{ \sim }y \), we have that \( {D}_{R}\left( x\right) = {D}_{R}\left( y\right) \) by Proposition 6.2.7 and, hence, that \( y \in {\mathcal{{DES}}}_{R}\left( i\right) \) . Also, we have a circular path\n\n\[ x \rightarrow {x}_{1} \rightarrow \cdots \rightarrow y \rightarrow {y}_{1} \rightarro... | Yes |
Theorem 6.5.2 If \( {\mathcal{C}}_{1} \) and \( {\mathcal{C}}_{2} \) are left cells of the same shape, then \( {\Gamma }_{{\mathcal{C}}_{1}} \cong \) \( {\Gamma }_{{\mathcal{C}}_{2}} \) (as labeled graphs). | Proof. Suppose that \( {\mathcal{C}}_{i} = \left\{ {x \in {S}_{n} : Q\left( x\right) = {Q}_{i}}\right\} \), for \( i = 1,2 \), where \( {Q}_{1} \) and \( {Q}_{2} \) are two tableaux of equal shape. We will show that the map \( \left( {P,{Q}_{1}}\right) \mapsto \left( {P,{Q}_{2}}\right) \) induces an isomorphism \( {\Ga... | Yes |
Theorem 6.5.3 \( K{L}_{\lambda } \cong {\rho }_{\lambda } \), for all \( \lambda \vdash n \) . | Proof. It follows from Theorems 6.3.8, 6.5.1, and 6.5.2 that\n\n\[ \n{\operatorname{Ind}}_{J}^{S}\left\lbrack 1\right\rbrack = {\bigoplus }_{\lambda \vdash n}\# \left\{ {Q \in {SY}{T}_{\lambda } : D\left( Q\right) \subseteq S \smallsetminus J}\right\} K{L}_{\lambda }.\n\]\n\nHence, Young's rule implies the conclusion. | Yes |
Proposition 6.6.1 Suppose \( \lambda = \left( {n - k,{1}^{k}}\right) \) . Then, \( {\Gamma }_{\lambda } \) is isomorphic (as a labeled graph) to the Hasse diagram of the poset \( L\left( {k, n - 1 - k}\right) \) . The label of a node equals the corresponding set. | Proof. The reading words of hook shape tableaux are \( \left\{ {x \in {S}_{n} : {D}_{R}\left( x\right) = }\right. \) \( \left\lbrack k\right\rbrack \} \) . These words are determined by the initial substring \( {x}_{1} > {x}_{2} > \cdots > \) \( {x}_{k} > {x}_{k + 1} = 1 \) . Hence, we get an identification \( x \leftr... | Yes |
Theorem 6.6.2 Fix \( \lambda \vdash n \) and identify the nodes of \( {\Gamma }_{\lambda } \) with \( {SY}{T}_{\lambda } \) . (i) Evacuation \( P \mapsto e\left( P\right) \) induces an order 2 automorphism of \( {\Gamma }_{\lambda } \) as an edge-labeled graph. The change of vertex labels is \( D\left( {e\left( P\right... | Proof. Part (i): Fix a right tableau \( Q \) so that our left cell \( \mathcal{C} \) of shape \( \lambda \) is realized as \( \mathcal{C} = \left\{ {x \in {S}_{n} : x \leftrightarrow \left( {P, Q}\right), P \in {SY}{T}_{\lambda }}\right\} \) . Let \( {\mathcal{C}}^{E} = \{ x \in \left. {{S}_{n} : x \leftrightarrow \lef... | Yes |
Lemma 7.1.2 Let \( J \subseteq S \) . Then,\n\n\[ W\left( q\right) = {W}^{J}\left( q\right) {W}_{J}\left( q\right) \] | Proof. This is an immediate consequence of Proposition 2.4.4. \( ▱ \) | No |
Proposition 7.1.3 Let \( I \subseteq J \subseteq S \) . Then,\n\n\[ \n{\mathcal{D}}_{I}^{J}\left( q\right) = \mathop{\sum }\limits_{{J \smallsetminus I \subseteq K \subseteq J}}{\left( -1\right) }^{\left| J \smallsetminus K\right| }{W}^{S \smallsetminus K}\left( q\right) .\n\] | Proof. It is clear from the definitions and equation (7.1) that\n\n\[ \n{W}^{S \smallsetminus K}\left( q\right) = \mathop{\sum }\limits_{{L \subseteq K}}{\mathcal{D}}_{L}\left( q\right) \n\]\n\nfor all \( K \subseteq S \) . Hence,\n\n\[ \n\mathop{\sum }\limits_{{J \smallsetminus I \subseteq K \subseteq J}}{\left( -1\ri... | Yes |
Corollary 7.1.4 We have the following:\n\n(i) If \( W \) is finite, then\n\n\[ \mathop{\sum }\limits_{{K \subseteq S}}\frac{{\left( -1\right) }^{\left| K\right| }}{{W}_{K}\left( q\right) } = \frac{{q}^{\ell \left( {w}_{0}\right) }}{W\left( q\right) } \]\n\n(ii) If \( W \) is infinite, then\n\n\[ \mathop{\sum }\limits_{... | Proof. Part (i) follows immediately by taking \( I = J = S \) in Proposition 7.1.3 and using Lemma 7.1.2. Part (ii) follows similarly using Proposition 2.3.1. \( ▱ \) | No |
Example 7.1.6 We illustrate this procedure with an example. Let \( \left( {W, S}\right) \) be a Coxeter system of type \( {F}_{4} \), and suppose that \( S = \left\{ {{s}_{1},\ldots ,{s}_{4}}\right\} \) and the \( {s}_{i} \) ’s are numbered as in Figure 3.6 (i.e., so that \( {\left( {s}_{1},{s}_{2}\right) }^{4} = e \) ... | \[ {W}^{J}\left( q\right) = 1 + q + {q}^{2} + {q}^{3} + 2\left( {{q}^{4} + \cdots + {q}^{11}}\right) + {q}^{12} + {q}^{13} + {q}^{14} + {q}^{15}. \] We then have to compute \( {W}_{J}\left( q\right) \) . However, \[ {W}_{J}\left( q\right) = {W}_{\left\{ {s}_{1},{s}_{2}\right\} }\left( q\right) {\left( {W}_{J}\right) }^... | Yes |
Proposition 7.1.7 Let \( \\left( {W, S}\\right) \) be an infinite Coxeter system. Then,\n\n\[ \n\\frac{1}{W\\left( q\\right) } = - \\mathop{\\sum }\\limits_{{K \\in \\mathcal{N}}}\\frac{{\\mu }_{\\mathcal{N}}\\left( {K, S}\\right) }{{W}_{K}\\left( q\\right) }\n\] | Proof. Let, for brevity, \( \\mathcal{N}\\overset{\\text{ def }}{ = }\\mathcal{N}\\left( {W, S}\\right) \) . Let \( I \\subseteq S, I \\notin \\mathcal{N} \) . Then, \( \\left| {W}_{I}\\right| = \\infty \) and, hence, we conclude from Corollary 7.1.4 that\n\n\[ \n\\mathop{\\sum }\\limits_{{K \\subseteq I}}\\frac{{\\lef... | Yes |
Example 7.1.9 We illustrate the difference, from a computational point of view, between Proposition 7.1.7 and Corollary 7.1.8 with an example. Let \( W = {\widetilde{A}}_{2} \) . Then, the nerve of \( \left( {W, S}\right) \) consists of all the proper subsets of \( S \) . So, \( \mathcal{N} \cup \{ S\} \) is a Boolean ... | \[ \frac{-1}{{\widetilde{A}}_{2}\left( q\right) } = \frac{-1}{1} + 3\frac{1}{{A}_{1}\left( q\right) } - 3\frac{1}{{A}_{2}\left( q\right) } \] \[ = \frac{-1}{1} + \frac{3}{{\left\lbrack 2\right\rbrack }_{q}} - \frac{3}{{\left\lbrack 2\right\rbrack }_{q}{\left\lbrack 3\right\rbrack }_{q}} \] \[ = \frac{\left( {q - 1}\rig... | Yes |
Theorem 7.1.10 Let \( \\left( {W, S}\\right) \) be an affine Coxeter system, and let \( {e}_{1},\\ldots ,{e}_{n} \) be the exponents of the corresponding finite group. Then,\n\n\[ \nW\\left( q\\right) = \\mathop{\\prod }\\limits_{{i = 1}}^{n}\\frac{{\\left\\lbrack {e}_{i} + 1\\right\\rbrack }_{q}}{1 - {q}^{{e}_{i}}}.\n... | Theorem 7.1.10 is due to Bott and a proof of it can be found in [78] or [295]. | No |
Theorem 7.2.1\n\n\\[ \nW\\left( {t;q}\\right) = \\mathop{\\sum }\\limits_{{J \\subseteq S}}{t}^{\\left| J\\right| }{\\left( 1 - t\\right) }^{\\left| S \\smallsetminus J\\right| }\\frac{W\\left( q\\right) }{{W}_{S \\smallsetminus J}\\left( q\\right) }.\n\\] | Proof. The key idea is to write the monomial \\( {t}^{d\\left( v\\right) } \\) in a clever way, namely\n\n\\[ \n{t}^{d\\left( v\\right) } = \\mathop{\\sum }\\limits_{{D\\left( v\\right) \\subseteq J \\subseteq S}}{t}^{\\left| J\\right| }{\\left( 1 - t\\right) }^{\\left| S \\smallsetminus J\\right| }.\n\\]\n\nThe result... | Yes |
Lemma 7.3.2 Let \( P \) and \( Q \) be two tableaux such that \( \operatorname{sh}\left( Q\right) \) extends \( \operatorname{sh}\left( P\right) \) . Then,\n\n\[ \n{j}^{P}\left( Q\right) = {v}_{Q}\left( P\right) \;\text{ and }\;{v}^{P}\left( Q\right) = {j}_{Q}\left( P\right) .\n\] | Proof. Let \( p = \left| P\right| \) and \( q = \left| Q\right| \) . We may clearly assume (by adding \( p \) to all the entries of \( Q \) ) that \( P \cup Q \) is standard. By repeated application of Lemma 7.3.1, we have that\n\n\[ \n{\left. {v}_{Q\left( i\right) }\left( P\right) \cup {j}_{Q\left( i\right) }\left( P\... | Yes |
Proposition 7.3.3 Let \( S, T \), and \( X \) be tableaux such that \( S \approx T \) and \( \operatorname{sh}\left( S\right) \) extends \( \operatorname{sh}\left( X\right) \) . Then, \[ {j}_{S}\left( X\right) = {j}_{T}\left( X\right) \] and \[ {v}_{S}\left( X\right) \approx {v}_{T}\left( X\right) \] | Proof. It follows imediately from the definition of dual equivalence that, with our hypotheses, \[ {v}^{X}\left( S\right) = {v}^{X}\left( T\right) \text{ and }{j}^{X}\left( S\right) \approx {j}^{X}\left( T\right) . \] So, the result follows from Lemma 7.3.2. \( ▱ \) | Yes |
Proposition 7.3.4 Let \( \lambda \smallsetminus \mu \) be a miniature shape. Then, \( \lambda \smallsetminus \mu \) is a brick if and only if, as a partially ordered set, \( \lambda \smallsetminus \mu \) is isomorphic to one of the posets in Figure 7.7. In particular, \( \lambda \smallsetminus \mu \) is a brick if and ... | Proof. We have already seen, in the examples following the definition of a brick, that the result holds if \( \lambda \smallsetminus \mu \) is isomorphic to \( {P}_{1} \) or \( {P}_{4} \) . Similarly, one can check it for \( {P}_{2} \) and \( {P}_{3} \) . If \( \lambda \smallsetminus \mu \) is not isomorphic to one of ... | Yes |
Theorem 7.3.5 Let \( S \) and \( T \) be two elementary dual equivalent tableaux of staircase shape, with \( n \) cells. Let \( \{ j, j + 1, j + 2\} \) be the entries of \( S \) and \( T \) involved in an elementary dual equivalence. Then, if \( - t ≢ j, j + 1\left( {\;\operatorname{mod}\;n}\right) \) , \( {p}^{t}\left... | Proof. If \( t = 1 \) and \( j + 2 < n \), or if \( t = - 1 \) and \( j > 1 \), then the result is clear from the definitions of dual equivalence and elementary dual equivalence. Thus, the result holds for any interval of allowed values of \( t \) if it holds for one element of the interval. However, the allowed values... | No |
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