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Proposition 10.5. Let \( x \) be an arbitrary point of \( C \) . Then we have\n\n\[ \mathop{\liminf }\limits_{\substack{{y \rightarrow x} \\ {y \in C} }}\frac{\lg \left| {G\left( x\right) - G\left( y\right) }\right| }{\lg \left| {x - y}\right| } = {s}_{c} \] | Proof. It suffices to show that\n\n\[ \mathop{\liminf }\limits_{\substack{{y \rightarrow x} \\ {y \in C} }}\frac{\lg \left| {G\left( x\right) - G\left( y\right) }\right| }{\lg \left| {x - y}\right| } \leq {s}_{c} \]\n\nChoosing \( \left| {x - y}\right| = \frac{2}{{3}^{n}} \) for \( n \rightarrow \infty \) we obtain the... | No |
Theorem 10.10. For \( {\mathcal{H}}^{{s}_{c}} \) almost all \( x \in C \) the logarithmic average density\n\n\[ \n{A}^{{s}_{c}}\left( x\right) \mathrel{\text{:=}} \mathop{\lim }\limits_{{T \rightarrow \infty }}\frac{1}{T}{\int }_{0}^{T}\frac{\widehat{G}\left( {x + {\mathrm{e}}^{-t}}\right) - \widehat{G}\left( {x - {\ma... | For the proof see [24, Theorem 6.6] and also [4]. | No |
Theorem 10.11 (Feng et al. [26]). 10.11.1. For all \( x \in C \) , | \[ {\Theta }_{ * }^{{s}_{c}}\left( x\right) = {\left( 4 - 6\tau \left( x\right) \right) }^{-{s}_{c}}. \] | Yes |
Theorem 2.3.4 (Mean Ergodic Theorem) Under the hypotheses in Theorem 2.2.5,\n\n\\[ \mathop{\sup }\limits_{{j \in \mathbb{S}}}\mathbb{E}\left\lbrack {\left( {\bar{T}}_{j}^{\left( n\right) } - {\left( \pi \right) }_{j}\right) }^{2}\right\rbrack \leq \frac{2\left( {M - 1}\right) }{n\epsilon }\;\text{ for all }n \geq 1. \\... | Proof Let \\( \\overline{\mathbf{f}} \\) be the column vector determined by the function \\( \\bar{f} = f - \\pi \\mathbf{f} \\) . Obviously,\n\n\\[ \frac{1}{n}\mathop{\sum }\limits_{{m = 0}}^{{n - 1}}f\left( {X}_{m}\right) - \\pi \\mathbf{f} = \frac{1}{n}\mathop{\sum }\limits_{{m = 0}}^{{n - 1}}\\bar{f}\left( {X}_{m}\... | Yes |
Theorem 2.3.6 For all \( m \in {\mathbb{Z}}^{ + } \) and \( \left( {i, j}\right) \in {\mathbb{S}}^{2} \) , \[ \mathbb{P}\left( {{\rho }_{j}^{\left( m\right) } < \infty \mid {X}_{0} = i}\right) = \mathbb{P}\left( {{\rho }_{j} < \infty \mid {X}_{0} = i}\right) \mathbb{P}{\left( {\rho }_{j} < \infty \mid {X}_{0} = j\right... | Proof To prove the first statement, we apply (2.1.11) and the monotone convergence theorem, Theorem 7.1.9, to justify \[ \mathbb{P}\left( {{\rho }_{j}^{\left( m\right) } < \infty \mid {X}_{0} = i}\right) \] \[ = \mathop{\sum }\limits_{{n = 1}}^{\infty }\mathbb{P}\left( {{\rho }_{j}^{\left( m - 1\right) } = n\& {\rho }_... | Yes |
Theorem 3.1.5 If \( \pi \) is a stationary probability for the transition probability \( \mathbf{P} \) on \( \mathbb{S} \) , then \( {\left( \mathbf{\pi }\right) }_{i} = 0 \) for all transient \( i \in \mathbb{S} \) . Furthermore, if \( i \) is recurrent and \( {\left( \mathbf{\pi }\right) }_{i} > 0 \) , then \( {\left... | Proof First observe that, for any \( n \geq 1 \) ,\n\n\[ n{\left( \mathbf{\pi }\right) }_{i} = \mathop{\sum }\limits_{{m = 0}}^{{n - 1}}{\left( \mathbf{\pi }{\mathbf{P}}^{m}\right) }_{i} = \mathop{\sum }\limits_{{j \in \mathbb{S}}}{\left( \mathbf{\pi }\right) }_{j}\mathbb{E}\left\lbrack {\mathop{\sum }\limits_{{m = 0}}... | Yes |
Theorem 3.1.6 If \( u \) is a non-negative function on \( \mathbb{S} \) with the property that \( {\left( \mathbf{{Pu}}\right) }_{i} \leq \) \( {\left( \mathbf{u}\right) }_{i} \) for all \( i \in \mathbb{S} \), then \( {\left( \mathbf{{Pu}}\right) }_{j} < {\left( \mathbf{u}\right) }_{j} \) for some \( j \in \mathbb{S} ... | Proof Set \( \mathbf{f} = \mathbf{u} - \mathbf{{Pu}} \), and note that, for all \( n \geq 1 \) ,\n\n\[ u\left( j\right) \geq {\left( \mathbf{u}\right) }_{j} - {\left( {\mathbf{P}}^{n}\mathbf{u}\right) }_{j} = \mathop{\sum }\limits_{{m = 0}}^{{n - 1}}\left( {{\left( {\mathbf{P}}^{m}\mathbf{u}\right) }_{j} - {\left( {\ma... | Yes |
Lemma 3.1.7 Assume that \( u : \mathbb{S} \rightarrow \mathbb{R} \) is bounded below and that \( \Gamma \) is a nonempty subset of \( \mathbb{S} \) . If \( {\left( \mathbf{{Pu}}\right) }_{i} \leq u\left( i\right) \) for all \( i \notin \Gamma \) and \( {\rho }_{\Gamma } \equiv \inf \left\{ {n \geq 1 : {X}_{n} \in \Gamm... | Proof Set \( {A}_{n} = \left\{ {{\rho }_{\Gamma } > n}\right\} \) . Then, \( {A}_{n} \) is measurable with respect to \( \left( {{X}_{0},\ldots ,{X}_{n}}\right) \) , and so, by (2.1.1), for any \( i \notin \Gamma \) ,\n\n\[ \mathbb{E}\left\lbrack {u\left( {X}_{\left( {n + 1}\right) \land {\rho }_{\Gamma }}\right) \mid ... | Yes |
Theorem 3.1.8 Assume that \( j \) is recurrent, and set \( C = \left\lbrack j\right\rbrack \) . If \( u : \mathbb{S} \rightarrow \lbrack 0,\infty ) \) is a bounded function and either \( u\left( i\right) = {\left( \mathbf{{Pu}}\right) }_{i} \) or \( u\left( j\right) \geq u\left( i\right) \geq {\left( \mathbf{{Pu}}\righ... | Proof In proving the first part, we will assume, without loss in generality, that \( C = \mathbb{S} \) . Now suppose that \( j \) is recurrent and that \( u\left( i\right) = {\left( \mathbf{{Pu}}\right) }_{i} \) for \( i \neq j \) . By applying Lemma 3.1.7 with \( \Gamma = \{ j\} \), we see that, for \( i \neq j \) ,\n... | Yes |
Lemma 3.1.9 If \( \mathbf{P} \) is irreducible on \( \mathbb{S} \), then, for any finite subset \( F \neq \mathbb{S} \) , \( \mathbb{P}\left( {{\rho }_{\mathbb{S} \smallsetminus F} < \infty \mid {X}_{0} = i}\right) = 1 \) for all \( i \in F. \) | Proof Set \( \tau = {\rho }_{\mathbb{S} \smallsetminus F} \) . By irreducibility, \( \mathbb{P}\left( {\tau < \infty \mid {X}_{0} = i}\right) > 0 \) for each \( i \in F \) . Hence, because \( F \) is finite, there exists a \( \theta \in \left( {0,1}\right) \) and an \( N \geq 1 \) such that \( \mathbb{P}\left( {\tau > ... | Yes |
Theorem 3.1.10 Assume that \( \mathbf{P} \) is irreducible on \( \mathbb{S} \), and let \( u : \mathbb{S} \rightarrow \lbrack 0,\infty ) \) be a function with the property that \( \{ k : u\left( k\right) \leq L\} \) is finite for each \( L \in \left( {0,\infty }\right) \) . If, for some \( j \in \mathbb{S},{\left( \mat... | Proof If \( \mathbb{S} \) is finite, then (cf., for example, Exercise 2.4.2) at least one state is recurrent, and therefore, by irreducibility, all are. Hence, we will assume that \( \mathbb{S} \) is infinite.\n\nGiven \( i \neq j \), set \( {F}_{L} = \{ k : u\left( k\right) \leq u\left( i\right) + u\left( j\right) + L... | Yes |
Theorem 3.1.13 Given \( \varnothing \neq S \subseteq \mathbb{Z} \) with \( S \neq \{ 0\} \) , \[ \gcd \left( S\right) \leq \min \{ \left| s\right| : s \in S \smallsetminus \{ 0\} \} \] and equality holds if and only if \( \{ \gcd \left( S\right) , - \gcd \left( S\right) \} \cap S \neq \varnothing \) . | Proof The first assertion needs no comment. To prove the second assertion, let \( \widehat{S} \) be the smallest subset of \( \mathbb{Z} \) which contains \( S \) and has the property that \( \left( {{s}_{1},{s}_{2}}\right) \in {\widehat{S}}^{2} \Rightarrow \) \( {s}_{1} \pm {s}_{2} \in \widehat{S} \) . As is easy to c... | Yes |
Corollary 3.1.16 Suppose that \( \mathbf{P} \) is an transition probability matrix on a finite state space \( \mathbb{S} \) . If there is an aperiodic state \( {j}_{0} \in \mathbb{S} \) such that \( i \rightarrow {j}_{0} \) for every \( i \in \mathbb{S} \) , then there exists an \( M \in {\mathbb{Z}}^{ + } \) and an \(... | Proof Because \( {j}_{0} \) is aperiodic, we know that there is an \( {M}_{0} \in \mathbb{N} \) such that \( {\left( {\mathbf{P}}^{n}\right) }_{{j}_{0}{j}_{0}} > 0 \) for all \( n \geq {M}_{0} \) . Further, because \( i \rightarrow {j}_{0} \), there exists an \( m\left( i\right) \in {\mathbb{Z}}^{ + } \) such that \( {... | Yes |
Theorem 3.2.1 Suppose that \( \mathbf{P} \) is a transition probability on a finite state space \( \mathbb{S} \) . Then there is at least one recurrent \( i \in \mathbb{S} \) . In addition, if \( j \) is transient, then \( j \rightarrow i \) for some recurrent \( i \) . Finally, suppose that \( \mathbf{P} \) admits pre... | Proof Begin by noting that, no matter where it starts, the chain can spend only a finite amount of time in the set \( \mathcal{T} \) of transient states. Indeed, if \( i \in \mathbb{S} \) and \( j \in \mathcal{T} \), then (cf. (2.3.7)) \( \mathbb{E}\left\lbrack {{T}_{j} \mid {X}_{0} = i}\right\rbrack < \infty \) and th... | Yes |
Lemma 3.2.2 If each row of \( \mathbf{A} \) sums to 0, then \( \operatorname{cof}{\left( \mathbf{A}\right) }_{ij} = \operatorname{cof}{\left( \mathbf{A}\right) }_{jj} \) for all \( 1 \leq i, j \leq N \) and\n\n\[ \mathop{\sum }\limits_{{i = 1}}^{N}\operatorname{cof}{\left( \mathbf{A}\right) }_{ii} = {\Pi }_{\mathbf{A}}... | Proof Given \( {i}_{1} \neq {i}_{2} \) and \( j \), let \( \mathbf{B} \) be the matrix whose \( \left( {j,{i}_{1}}\right) \) st entry is 1, whose \( \left( {j,{i}_{2}}\right) \) th entry is -1, and whose other entries are all 0 . Then, for each \( t \in \mathbb{R} \), all the rows of \( \mathbf{A} + t\mathbf{B} \) sum ... | Yes |
Theorem 3.2.6 Let \( \mathbf{P} \) be a transition probability on a finite state space \( \mathbb{S} \), and assume that \( \mathbf{\pi } \) is the one and only stationary probability for \( \mathbf{P} \). Then 0 is a simple eigenvalue of \( \mathbf{I} - \mathbf{P} \) and, if \( {\Pi }_{\mathbf{I} - \mathbf{P}} \) is t... | Proof To prove (3.2.7), take \( \mathbf{A} = \mathbf{I} - \mathbf{P} \). Clearly the rows of \( \mathbf{A} \) each sum to 0, and, by Lemma 3.2.5, 0 is a simple eigenvalue of \( \mathbf{A} \). In addition, again by Lemma 3.2.5, the \( \pi \) in (3.2.7) is a probability vector and, by Theorem 3.2.3, it is the one and onl... | Yes |
Corollary 3.2.8 Let \( \\mathbf{P} \) and \( \\pi \) be as in Theorem 3.2.6, and denote by \( \\mathcal{T} \) the set of transient states. Then \( {\\left( \\pi \\right) }_{i} = 0 \) for \( i \\in \\mathcal{T} \) and \[ {\\left( \\mathbf{\\pi }\\right) }_{i} = \\frac{\\det \\left( {\\left( \\mathbf{I} - \\mathbf{P}\\ri... | Proof First notice that \( {\\mathbf{P}}^{\\mathcal{T}} \) is a transition probability on \( \\mathbb{S} \\smallsetminus \\mathcal{T} \) and that \( {\\mathbf{\\pi }}^{\\mathcal{T}} = {\\mathbf{\\pi }}^{\\mathcal{T}}{\\mathbf{P}}^{\\mathcal{T}} \) if \( {\\mathbf{\\pi }}^{\\mathcal{T}} \) is the restriction of \( \\mat... | Yes |
Theorem 3.2.9 Let \( \mathbf{P} \) and \( \mathbb{S} \) be as in Theorem 3.2.6. If \( i \) is recurrent and \( i \in \Delta \subsetneq \mathbb{S} \) , then \( {\left( \mathbf{I} - \mathbf{P}\right) }^{\Delta } \) is invertible and\n\n\[{\left( {\left( {\left( \mathbf{I} - \mathbf{P}\right) }^{\Delta }\right) }^{-1}\rig... | Proof First observe that\n\n\[ \mathbb{P}\left( {{X}_{m} = {j}_{m}\text{ for }0 \leq m \leq n\& {\zeta }^{\Delta } > n \mid {X}_{0} = {j}_{0}}\right)\]\n\n\[= {\mathbf{1}}_{\Delta \mathbf{C}}\left( {j}_{0}\right) \mathop{\prod }\limits_{{m = 1}}^{n}{\mathbf{1}}_{\Delta \mathbf{C}}\left( {j}_{m}\right) {\left( \mathbf{P... | Yes |
Theorem 3.3.4 (Wilson) For any spanning tree \( T \) and any vertex \( v \in V \) ,\n\n\[ \n\mathcal{P}\left( T\right) = \frac{1}{\det \left( {\left( \mathcal{D} - \mathcal{A}\right) }^{\{ v\} }\right) }.\n\]\n\nIn particular, \( \mathcal{P}\left( T\right) \) is the same for all spanning trees \( T \) . | Proof Choose an ordering \( \left( {{v}_{1},\ldots ,{v}_{N}}\right) \) with \( {v}_{2} = v \), let \( \left( {{P}_{1},\ldots ,{P}_{L}}\right) \) be the Wilson run determined by \( T \) that respects this ordering, define \( {\Delta }_{\ell } \) as in (3.3.1) for \( 1 \leq \ell \leq L \), and set \( {\Delta }_{L + 1} = ... | Yes |
Theorem 4.1.10 Stat(P) is a convex subset of \( {\mathbb{R}}^{\mathbb{S}} \) . Moreover, \( \operatorname{Stat}\left( \mathbf{P}\right) \neq \varnothing \) if and only if there is at least one positive recurrent state \( j \in \mathbb{S} \) . In fact, for any \( \mathbf{\mu } \in \operatorname{Stat}\left( \mathbf{P}\ri... | Proof The only statements not already covered are the characterization of the extreme points of \( \operatorname{Stat}\left( \mathbf{P}\right) \) and the final assertion in the case when \( j \) is recurrent.\n\nIn view of (4.1.8), the final assertion when \( j \) is recurrent comes down to showing that if \( j \) is p... | Yes |
For all \( \left( {i, j}\right) ,{\overline{\lim }}_{n \rightarrow \infty }{\left( {\mathbf{A}}_{n}\right) }_{ij} \leq e{\pi }_{jj} \) . In addition, for any \( j \) and any subsequence \( \left\{ {{n}_{\ell } : \ell \geq 0}\right\} \subseteq \mathbb{N} \) , \[ \mathop{\lim }\limits_{{\ell \rightarrow \infty }}{\left( ... | Proof To prove the first part, observe that \[ {\left( {\mathbf{A}}_{n}\right) }_{ij} \leq \frac{1}{n}{\left( 1 - \frac{1}{n}\right) }^{-n}\mathop{\sum }\limits_{{m = 0}}^{{n - 1}}{\left( 1 - \frac{1}{n}\right) }^{m}{\left( {\mathbf{P}}^{m}\right) }_{ij} \leq {\left( 1 - \frac{1}{n}\right) }^{-n}{\left( \mathbf{R}\left... | Yes |
Theorem 4.1.14 Let \( C \) a communicating class of positive recurrent states. If \( \mathbb{P}\left( {{X}_{0} \in C}\right) = 1 \), then | Proof Since \( \mathbb{P}\left( {{X}_{m} \in C}\right. \) for all \( \left. {m \in \mathbb{N}}\right) = 1 \), without loss in generality we may and will assume that \( C \) is the whole state space. In keeping with this assumption, we will set \( \pi = {\pi }^{C} \). Next note that if \( {\mu }_{i} = \mathbb{P}\left( {... | No |
Lemma 5.2.5 A bounded function \( w : \lbrack 0,\infty ) \times \mathbb{S} \rightarrow \mathbb{R} \) that is continuously differentiable with respect to \( t \geq 0 \) satisfies\n\n\[ \dot{w}\left( {t, j}\right) = {\left( \mathbf{Q}\mathbf{w}\left( t\right) \right) }_{j}, \]\n\nwhere \( \mathbf{w}\left( t\right) \) is ... | Proof Let \( \mathbf{w} \) be a row vector with \( \parallel \mathbf{w}{\parallel }_{\mathbf{u}} < \infty \), and define \( w\left( {t, j}\right) = {\left( \mathbf{{wP}}\left( t\right) \right) }_{j} \) for \( \left( {t, j}\right) \in \lbrack 0,\infty ) \times \mathbb{S} \) . Then \( \left| {w\left( {t, j}\right) }\righ... | Yes |
Theorem 5.2.13 Let \( \{ X\left( t\right) : t \geq 0\} \) be a Markov process generated by \( \mathbf{Q} \) and let \( \mathfrak{R} \) and \( \mathbf{P} \) be given by (5.2.12). Set \( {J}_{0} = 0 \), \[ {J}_{n} = \inf \left\{ {t > {J}_{n - 1} : X\left( t\right) \neq X\left( {J}_{n - 1}\right) }\right\} \;\text{ for }n... | Proof Refer to Sect. 5.2.2 and let \( \{ \widetilde{X}\left( t\right) : t \geq 0\} \) be the process in Theorem 5.2.10 when \( \mathbf{\mu } \) is the distribution of \( X\left( 0\right) \) . Because it has the same distribution as \( \{ X\left( t\right) \) : \( t \geq 0\} \), it suffices to prove that \( \{ \widetilde... | Yes |
Lemma 5.3.2 \( \mathbb{P}\left( {\widetilde{\mathfrak{e}} = {\widetilde{J}}_{\infty }}\right) = 1 \) and so, with probability \( 1,\widetilde{X}\left( t\right) \) is well-defined for \( t \in \lbrack 0,\widetilde{\mathfrak{e}}) \) . | Proof Because \( \left\{ {\widetilde{\mathfrak{e}} \neq {\widetilde{J}}_{\infty }}\right\} \) can be written as the union of the sets\n\n\[ \left\{ {\widetilde{\mathfrak{e}} > T \geq {\widetilde{J}}_{\infty }}\right\} \cup \left\{ {{\widetilde{J}}_{\infty } > T \geq \widetilde{\mathfrak{e}}}\right\} \]\n\nas \( T \) ru... | Yes |
Lemma 5.3.3 If \( \mathbb{P}\left( {\widetilde{\mathfrak{e}} = \infty \mid \widetilde{X}\left( 0\right) = i}\right) = 1 \) for all \( i \in \mathbb{S} \) and\n\n\[ \n{\left( \mathbf{P}\left( t\right) \right) }_{ij} \equiv \mathbb{P}\left( {\widetilde{X}\left( t\right) = j \mid \widetilde{X}\left( 0\right) = i}\right) ,... | Proof First note that\n\n\[ \n\mathbb{P}\left( {\mathfrak{e} = \infty }\right) = \mathop{\sum }\limits_{{i \in \mathbb{S}}}{\left( \mathbf{\mu }\right) }_{i}\mathbb{P}\left( {\mathfrak{e} = \infty \mid X\left( 0\right) = i}\right) = 1,\n\]\n\nand therefore that \( \mathop{\lim }\limits_{{N \rightarrow \infty }}\mathbb{... | Yes |
Theorem 5.3.6 Let \( \mathbf{Q} \) be a \( Q \) -matrix and, for \( N \geq 1 \), determine the \( Q \) -matrix \( {\mathbf{Q}}^{\left( N\right) } \) by \( {\left( {\mathbf{Q}}^{\left( N\right) }\right) }_{ij} = {\mathbf{1}}_{{F}_{N}}\left( i\right) {\left( \mathbf{Q}\right) }_{ij} \) . Given a point \( \Delta \) not in... | Proof Let \( \mathfrak{R} \) and \( \mathbf{P} \) be the canonical rates and transition probability determined by \( \mathbf{Q} \), and define \( \left\{ {\widetilde{X}\left( t\right) : t \in \left\lbrack {0,{\widetilde{J}}_{\infty }}\right) }\right\} \) accordingly for \( \mathfrak{R} \) and \( \mathbf{P} \) with \( \... | Yes |
Corollary 5.3.7 Assume that explosion does not occur for any \( i \in \mathbb{S} \) . Then, for each probability vector \( \mathbf{\mu } \) on \( \mathbb{S} \) there exists a right continuous, piecewise constant Markov process \( \{ X\left( t\right) : t \geq 0\} \) with initial distribution \( \mathbf{\mu } \) such tha... | Proof For each \( i \in \mathbb{S} \) let \( \left\{ {{X}_{i}\left( t\right) : t \geq 0}\right\} \) be a Markov process generated by \( \mathbf{Q} \) starting from \( i \), let \( {X}_{0} \) be a random variable with distribution \( \mathbf{\mu } \) which is independent of \( \sigma \left( \left\{ {{X}_{i}\left( t\righ... | Yes |
Theorem 5.3.9 If \( \mathbf{P} \) is a transition probability matrix and if \( i \in \mathbb{S} \) is \( \mathbf{P} \)-recurrent, then for every choice of rates \( \Re ,\mathbb{P}\left( {\mathfrak{e} = \infty }\right) = 1 \) for the process in Theorem 5.3.6 corresponding to the \( Q \)-matrix determined by \( \mathfrak... | Proof By Lemma 5.3.2, what we must show is that \( \mathbb{P}\left( {{\widetilde{J}}_{\infty } = \infty }\right) = 1 \). Equivalently, if \( \left\{ {{\widetilde{X}}_{n} : n \geq 0}\right\} \) is a Markov chain with transition probability \( \mathbf{P} \) with \( {\widetilde{X}}_{0} = i \) and if \( \left\{ {{\widetild... | Yes |
Theorem 5.3.10 If there exists a non-negative function \( u \) on \( \mathbb{S} \) with the properties that \( {U}_{N} \equiv \mathop{\inf }\limits_{{j \notin {F}_{N}}}u\left( j\right) \rightarrow \infty \) as \( N \rightarrow \infty \) and, for some \( \alpha \in \lbrack 0,\infty ) \) ,\n\n\[ \mathop{\sum }\limits_{{j... | Proof To prove the first part, for each \( N \geq 1 \), set \( {u}^{\left( N\right) }\left( j\right) = u\left( j\right) \) when \( j \in {F}_{N} \) and \( {u}^{\left( N\right) }\left( j\right) = {U}_{N} \) when \( j \notin {F}_{N} \) . It is an easy matter to check that if \( {\mathbf{Q}}^{\left( N\right) } = \) \( {\m... | Yes |
Theorem 5.4.3 For any given state \( i \in \mathbb{S} \), the following are equivalent:\n\n(1) \( i \) is \( \mathbf{Q} \)-recurrent.\n\n(2) There is a \( t \in \left( {0,\infty }\right) \) such that \( i \) is recurrent relative to the transition probability \( \mathbf{P}\left( t\right) \).\n\n(3) \( i \) is recurrent... | Proof We will prove this equivalence by checking that the same equivalence holds when \ | No |
Theorem 5.4.6 For each \( j \in \mathbb{S} \)\n\n\[ \n{\widehat{\pi }}_{jj} \equiv \mathop{\lim }\limits_{{t \rightarrow \infty }}{\left( \mathbf{P}\left( t\right) \right) }_{jj}\;\text{ exists }\n\]\n\nand\n\n\[ \n\mathop{\lim }\limits_{{t \rightarrow \infty }}{\left( \mathbf{P}\left( t\right) \right) }_{ij} = {\wideh... | Proof We begin with the following continuous-time version of the renewal equation (cf. (4.1.6)):\n\n\[ \n{\left( \mathbf{P}\left( t\right) \right) }_{ij} = {e}^{-t{R}_{i}}{\delta }_{i, j} + \mathbb{E}\left\lbrack {{\left( \mathbf{P}\left( t - {\sigma }_{j}\right) \right) }_{jj},{\sigma }_{j} \leq t \mid X\left( 0\right... | Yes |
Corollary 5.4.8 (Mean Ergodic Theorem) Assume that \( j \) is \( \mathbf{Q} \) -positive recurrent and that \( \mathbb{P}\left( {X\left( 0\right) \overset{\mathbf{Q}}{ \leftrightarrow }j}\right) = 1 \) . Then, | \[ \mathop{\lim }\limits_{{T \rightarrow \infty }}\mathbb{E}\left\lbrack {\left( \frac{1}{T}{\int }_{0}^{T}{\mathbf{1}}_{\{ j\} }\left( X\left( t\right) \right) dt - {\widehat{\pi }}_{jj}\right) }^{2}\right\rbrack = 0. \] Proof The proof is really just an obvious transcription to the continuous setting of the argument ... | Yes |
Theorem 5.4.11 Assume that \( \mathbb{S} \) is finite and that there is only one stationary probability \( \widehat{\pi } \) for \( t \rightsquigarrow \mathbf{P}\left( t\right) \) . Then 0 is a simple eigenvalue of \( \mathbf{Q} \) and\n\n\[ \n{\left( \widehat{\mathbf{\pi }}\right) }_{i} = \frac{\det \left( {\left( -Q\... | Proof The argument is essentially the same as the one for chains. To see that 0 is a simple eigenvalue, suppose not. Then there exists a column vector \( \mathbf{v} \neq \mathbf{0} \) such that \( \mathbf{Q}\mathbf{v} = \mathbf{0} \) and \( \widehat{\pi }\mathbf{v} = 0 \) . By (5.4.10), \( \mathbf{v} = \mathbf{P}\left(... | Yes |
Theorem 6.1.14 If \( \mathbf{P} \) is an irreducible transition probability for which there is a reversible initial distribution, which is necessarily \( \mathbf{\pi } \), then the period of \( \mathbf{P} \) is either 1 or 2. Moreover, the period is 2 if and only if there exists an \( f \in {L}^{2}\left( \mathbf{\pi }\... | Proof We begin by showing that the period \( d \) must be less than or equal to 2 . To this end, remember that, because of irreducibility, \( {\left( \mathbf{\pi }\right) }_{i} > 0 \) for all \( i \) ’s. Hence, the detailed balance condition,(6.1.1), implies that \( {\left( \mathbf{P}\right) }_{ij} > 0 \Leftrightarrow ... | Yes |
Lemma 6.3.9 If \( 0 < s < t \), then for any \( f \in {L}^{2}\left( \widehat{\pi }\right) \)\n\n\[ \frac{\parallel f{\parallel }_{2,\widehat{\mathbf{\pi }}}^{2}}{s} \geq \frac{\parallel \mathbf{P}\left( s\right) f{\parallel }_{2,\widehat{\mathbf{\pi }}}^{2} - \parallel \mathbf{P}\left( t\right) f{\parallel }_{2,\wideha... | Proof Set \( \psi \left( t\right) = \parallel \mathbf{P}\left( t\right) f{\parallel }_{2,\widehat{\pi }}^{2} \) . We know that \( \psi \) is a continuous, non-increasing, non-negative, convex function. Hence, by part (a) of Exercise 5.5.2,\n\n\[ \frac{\psi \left( 0\right) }{s} \geq \frac{\psi \left( 0\right) - \psi \le... | Yes |
Theorem 6.4.11 Assume that \( \mathbb{S} \) is finite and that \( \mathbf{Q}\left( \beta \right) \) is given by (6.4.7). Set \( \mathfrak{m} = \mathop{\min }\limits_{{i \in \mathbb{S}}}H\left( i\right) \) and \( {\mathbb{S}}_{0} = \{ i : H\left( i\right) = \mathfrak{m}\} \), and let \( \mathfrak{e} \) be the minimum va... | Proof Because neither \( \mathbf{\gamma }\left( \beta \right) \) nor \( \mathbf{Q}\left( \beta \right) \) is changed if \( H \) is replaced by \( H - \mathfrak{m} \) whereas \( \mathfrak{e} \) changes to \( \mathfrak{e} + \mathfrak{m} \), we may and will assume that \( \mathfrak{m} = 0 \). Choose a collection \( \mathc... | Yes |
Theorem 7.1.6 Suppose that \( \left( {\Omega ,\mathcal{F}}\right) \) is a measurable space and that \( \mathcal{C} \subseteq \mathcal{F} \) includes \( \Omega \) and is closed under intersection (i.e., \( A \cap B \in \mathcal{C} \) whenever \( A, B \in \mathcal{C} \) ). If \( \mu \) and \( v \) are a pair of finite me... | Proof We will say that \( \mathcal{S} \subseteq \mathcal{F} \) is good if\n\n(i) \( A, B \in \mathcal{S} \) and \( A \subseteq B \Rightarrow B \smallsetminus A \in \mathcal{S} \) .\n\n(ii) \( A, B \in \mathcal{S} \) and \( A \cap B = \varnothing \Rightarrow A \cup B \in \mathcal{S} \) .\n\n(iii) \( {\left\{ {A}_{n}\rig... | Yes |
Theorem 7.1.10 (Fatou’s Lemmas) Given any sequence \( {\left\{ {f}_{n}\right\} }_{1}^{\infty } \) of measurable functions, all of which dominate some fixed integrable function \( g \) , | \[ \mathop{\lim }\limits_{{n \rightarrow \infty }}\int {f}_{n}{d\mu } \geq \int \mathop{\lim }\limits_{{n \rightarrow \infty }}{f}_{n}{d\mu } \] | Yes |
Theorem 7.1.15 (Fubini) Let \( \\left( {{\\Omega }_{1},{\\mathcal{F}}_{1},{\\mu }_{1}}\\right) \) and \( \\left( {{\\Omega }_{2},{\\mathcal{F}}_{2},{\\mu }_{2}}\\right) \) be a pair of \( \\sigma \) -finite measure spaces, and set \( \\Omega = {\\Omega }_{1} \\times {\\Omega }_{2} \) and \( \\mathcal{F} = {\\mathcal{F}... | \[ {\\omega }_{1} \\rightsquigarrow {\\int }_{{\\Omega }_{2}}f\\left( {{\\omega }_{1},{\\omega }_{2}}\\right) {\\mu }_{2}\\left( {d{\\omega }_{2}}\\right) \\;\\text{ and }\\;{\\omega }_{2} \\rightsquigarrow {\\int }_{{\\Omega }_{1}}f\\left( {{\\omega }_{1},{\\omega }_{2}}\\right) {\\mu }_{1}\\left( {d{\\omega }_{1}}\\r... | Yes |
Lemma 7.3.1 Given any family \( {\left\{ {B}_{m}\right\} }_{1}^{\infty } \) of mutually independent, \( \{ 0,1\} \) -valued Bernoulli random variables satisfying \( \mathbb{P}\left( {{B}_{m} = 0}\right) = \frac{1}{2} = \mathbb{P}\left( {{B}_{m} = 1}\right) \) for all \( m \in {\mathbb{Z}}^{ + } \) , set \( U = \mathop{... | Proof Given \( N \geq 1 \) and \( 0 \leq n < {2}^{N} \), we want to show that\n\n\[ \mathbb{P}\left( {n{2}^{-N} < U \leq \left( {n + 1}\right) {2}^{-N}}\right) = {2}^{-N}. \]\n\n\( \left( *\right) \)\n\nTo this end, note that \( n{2}^{-N} < U \leq \left( {n + 1}\right) {2}^{-N} \) if and only if \( \mathop{\sum }\limit... | Yes |
The path  is the Coxeter graph of the symmetric group \( {S}_{n} \) with respect to the generating system of adjacent transpositions \( {s}_{i} = \left( {i, i + 1}\right) ,1 \leq i \leq n - 1 \). | This is proved in Proposition 1.5.4. | No |
Let \( {L}_{1} \) and \( {L}_{2} \) be straight lines through the origin of the Euclidean plane \( {\mathbb{E}}^{2} \). Assume that the angle between them is \( \frac{\pi }{m} \), for some integer \( m \geq 2 \). Let \( {r}_{1} \) be the orthogonal reflection through \( {L}_{1} \), and similarly for \( {r}_{2} \). Then... | Let \( {G}_{m} \) be the group generated by \( {r}_{1} \) and \( {r}_{2} \). Simple geometric considerations show that \( {G}_{m} \) consists of the \( m \) rotations of the plane through angles \( \frac{2\pi k}{m},0 \leq k < m \), and these \( m \) rotations followed by the reflection \( {r}_{1} \). Hence, \( \left| {... | Yes |
Definition. A pair \( B, N \) of subgroups of a group \( G \) is called a \( {BN} \) -pair (or Tits system) if the following hold:\n\n(1) \( B \cup N \) generates \( G \), and \( B \cap N \) is normal in \( N \).\n\n(2) \( W\overset{\text{ def }}{ = }N/\left( {B \cap N}\right) \) is generated by some set \( S \) of inv... | It can be shown to follow from these axioms that the set \( S \) is uniquely determined and that the pair \( \left( {W, S}\right) \) is a Coxeter system. The group \( W \) is called the Weyl group and the number \( \left| S\right| \) is the rank of the BN-pair \( \left( {G;B, N}\right) \) . | Yes |
Lemma 1.3.1 If \( w = {s}_{1}{s}_{2}\ldots {s}_{k} \), with \( k \) minimal, then \( {t}_{i} \neq {t}_{j} \) for all \( 1 \leq i < j \leq k \) . | Proof. If \( {t}_{i} = {t}_{j} \) for some \( i < j \) then \( w = {t}_{i}{t}_{j}{s}_{1}{s}_{2}\ldots {s}_{k} = \) \( {s}_{1}\ldots {\widehat{s}}_{i}\ldots {\widehat{s}}_{j}\ldots {s}_{k} \) (i.e., \( {s}_{i} \) and \( {s}_{j} \) deleted), which contradicts the mini-mality of \( k \) . \( ▱ \) | Yes |
Proposition 1.4.2 For all \( u, w \in W \) :\n\n(i) \( \varepsilon \left( w\right) = {\left( -1\right) }^{\ell \left( w\right) } \),\n\n(ii) \( \ell \left( {uw}\right) \equiv \ell \left( u\right) + \ell \left( w\right) \left( {\;\operatorname{mod}\;2}\right) \),\n\n(iii) \( \ell \left( {sw}\right) = \ell \left( w\right... | Proof. Parts (i) - (iii) follow from Lemma 1.4.1. We leave the rest as exercises. \( ▱ \) | No |
Theorem 1.4.3 (Strong Exchange Property) Suppose \( w = {s}_{1}{s}_{2}\ldots {s}_{k} \) \( \left( {{s}_{i} \in S}\right) \) and \( t \in T \) . If \( \ell \left( {tw}\right) < \ell \left( w\right) \), then \( {tw} = {s}_{1}\ldots {\widehat{s}}_{i}\ldots {s}_{k} \) for some \( i \in \left\lbrack k\right\rbrack \) . | Proof. Recall the number \( \eta \left( {w;t}\right) \in \{ + 1, - 1\} \) defined in definition (1.17). We prove the equivalence of these two conditions:\n\n(a) \( \ell \left( {tw}\right) < \ell \left( w\right) \), \n\n(b) \( \eta \left( {w;t}\right) = - 1 \). \n\nFirst, assume that \( \eta \left( {w;t}\right) = - 1 \)... | Yes |
Corollary 1.4.5 \( \;\left| {{T}_{L}\left( w\right) }\right| = \ell \left( w\right) \) . | Proof. Let \( w = {s}_{1}{s}_{2}\ldots {s}_{k}, k = \ell \left( w\right) \) . Then, \( {T}_{L}\left( w\right) = \left\{ {{s}_{1}{s}_{2}\ldots {s}_{i}\ldots {s}_{2}{s}_{1}}\right. \) : \( 1 \leq i \leq k\} \) by Corollary 1.4.4, and these elements are all distinct by Lemma 1.3.1. \( ▱ \) | Yes |
Corollary 1.4.6 For all \( s \in S \) and \( w \in W \), the following hold:\n\n(i) \( s \in {D}_{L}\left( w\right) \) if and only if some reduced expression for \( w \) begins with the letter \( s \) .\n\n(ii) \( s \in {D}_{R}\left( w\right) \) if and only if some reduced expression for \( w \) ends with the letter \(... | Proof. The \ | No |
Proposition 1.4.7 (Deletion Property) If \( w = {s}_{1}{s}_{2}\ldots {s}_{k} \) and \( \ell \left( w\right) < \) \( k \), then \( w = {s}_{1}\ldots {\widehat{s}}_{i}\ldots {\widehat{s}}_{j}\ldots {s}_{k} \) for some \( 1 \leq i < j \leq k \) . | Proof. Choose \( i \) maximal so that \( {s}_{i}{s}_{i + 1}\ldots {s}_{k} \) is not reduced. Then, \( \ell \left( {{s}_{i}{s}_{i + 1}\ldots {s}_{k}}\right) < \ell \left( {{s}_{i + 1}\ldots {s}_{k}}\right) \) and hence, by Theorem 1.4.3,\n\n\[ {s}_{i}{s}_{i + 1}\ldots {s}_{k} = {s}_{i + 1}\ldots {\widehat{s}}_{j}\ldots ... | Yes |
Corollary 1.4.8 (i) Any expression \( w = {s}_{1}{s}_{2}\ldots {s}_{k} \) contains a reduced expression for \( w \) as a subword, obtainable by deleting an even number of letters. | Proof. Part (i) is a direct consequence of the deletion property. | No |
Theorem 1.5.1 Let \( W \) be a group and \( S \) a set of generators of order 2. Then the following are equivalent:\n\n(i) \( \left( {W, S}\right) \) is a Coxeter system.\n\n(ii) \( \left( {W, S}\right) \) has the Exchange Property.\n\n(iii) \( \left( {W, S}\right) \) has the Deletion Property. | Proof. (i) \( \Rightarrow \) (ii) This is a special case of Theorem 1.4.3.\n\n(ii) \( \Rightarrow \) (iii) The proof of Proposition 1.4.7 goes through to prove this implication, even if \( \left( {W, S}\right) \) is not (a priori) a Coxeter system.\n\n(iii) \( \Rightarrow \) (ii) Suppose \( \ell \left( {s{s}_{1}\ldots ... | Yes |
Proposition 1.5.2 Let \( x \in {S}_{n} \) . Then,\n\n\[{\ell }_{A}\left( x\right) = \operatorname{inv}\left( x\right)\] | Proof. Since \( \operatorname{inv}\left( e\right) = {\ell }_{A}\left( e\right) = 0 \), relation (1.26) implies that \( \operatorname{inv}\left( x\right) \leq \) \( {\ell }_{A}\left( x\right) \) . The opposite inequality will be proved by induction on \( \operatorname{inv}\left( x\right) \) .\n\nIf \( \operatorname{inv}... | Yes |
Proposition 1.5.3 Let \( x \in {S}_{n} \) . Then,\n\n\[ \n{D}_{R}\left( x\right) = \left\{ {{s}_{i} \in S : x\left( i\right) > x\left( {i + 1}\right) }\right\} .\n\] | Proof. By the definitions and Proposition 1.5.2, we have that\n\n\[ \n{D}_{R}\left( x\right) = \{ s \in S : \operatorname{inv}\left( {xs}\right) < \operatorname{inv}\left( x\right) \}\n\]\n\nso (1.28) follows from (1.26). \( ▱ \) | No |
Proposition 1.5.4 \( \left( {{S}_{n}, S}\right) \) is a Coxeter system of type \( {A}_{n - 1} \) . | Proof. We show that the pair \( \left( {{S}_{n}, S}\right) \) has the Exchange Property (in its \ | No |
Example 2.1.2 Consider the dihedral group \( {I}_{2}\left( 4\right) \cong {B}_{2} \) with Coxeter graph | Then, \( T = \{ a, b,{aba},{bab}\} \) and the group has the following diagram under Bruhat order:\n\n\n\nFigure 2.1. Bruhat order of \( {B}_{2} \) .\n\nTo obtain the Bruhat graph of \( {B}_{2} \), direct all edges of F... | Yes |
How are these closed cells arranged? The elegant answer is | \[ \overline{{C}_{u}} \subseteq \overline{{C}_{w}}\; \Leftrightarrow \;u \leq w \] that is, the combinatorial pattern of inclusion of Bruhat cells determines Bruhat order on \( {S}_{n} \) . | Yes |
Lemma 2.1.4 Let \( x, y \in {S}_{n} \). Then, \( x \) is covered by \( y \) in Bruhat order if and only if \( y = x \cdot \left( {a, b}\right) \) for some \( a < b \) such that \( x\left( a\right) < x\left( b\right) \) and there does not exist any \( c \) such that \( a < c < b, x\left( a\right) < x\left( c\right) < x\... | Proof. If \( y = x \cdot \left( {a, b}\right) \) with the stated properties, then \( \operatorname{inv}\left( y\right) = \operatorname{inv}\left( x\right) + 1 \); hence, we have a Bruhat covering. Suppose conversely that \( y = x \cdot \left( {a, b}\right) \), \( a < b \), and \( \operatorname{inv}\left( y\right) > \op... | Yes |
Lemma 2.2.1 For \( u, w \in W, u \neq w \), let \( w = {s}_{1}{s}_{2}\ldots {s}_{q} \) be reduced, and suppose that some reduced expression for \( u \) is a subword of \( {s}_{1}{s}_{2}\ldots {s}_{q} \) . Then, there exists \( v \in W \) such that the following hold:\n\n(i) \( v > u \) .\n\n(ii) \( \ell \left( v\right)... | Proof. Of all reduced subword expressions\n\n\[ u = {s}_{1}\ldots {\widehat{s}}_{{i}_{1}}\ldots {\widehat{s}}_{{i}_{k}}\ldots {s}_{q},\;1 \leq {i}_{1} < \cdots < {i}_{k} \leq q, \]\n\nchoose one such that \( {i}_{k} \) is minimal. Let\n\n\[ t = {s}_{q}{s}_{q - 1}\ldots {s}_{{i}_{k}}\ldots {s}_{q - 1}{s}_{q}. \]\n\nThen... | Yes |
Theorem 2.2.2 (Subword Property) Let \( w = {s}_{1}{s}_{2}\ldots {s}_{q} \) be a reduced expression. Then,\n\n\( u \leq w\; \Leftrightarrow \; \) there exists a reduced expression\n\n\[ u = {s}_{{i}_{1}}{s}_{{i}_{2}}\ldots {s}_{{i}_{k}},\;1 \leq {i}_{1} < \ldots < {i}_{k} \leq q. \] | Proof. \( \left( \Rightarrow \right) \) Suppose that \( u = {x}_{0}\overset{{t}_{1}}{ \rightarrow }{x}_{1}\overset{{t}_{2}}{ \rightarrow }\cdots \overset{{t}_{m}}{ \rightarrow }{x}_{m} = w \) . Then, \( {x}_{m - 1} = w{t}_{m} = {s}_{1}\ldots {\widehat{s}}_{i}\ldots {s}_{q} \) for some \( i \) by the Strong Exchange Pro... | Yes |
Corollary 2.2.4 Bruhat intervals \( \left\lbrack {u, w}\right\rbrack \) are finite (even if \( S \) is infinite). In fact, \( \operatorname{card}\left\lbrack {u, w}\right\rbrack \leq {2}^{\ell \left( w\right) } \) . | Proof. There are \( {2}^{\ell \left( w\right) } \) subwords of any reduced expression for \( w \), and there is an injective map from \( \left\lbrack {u, w}\right\rbrack \) into the set of those subwords. \( ▱ \) | Yes |
Corollary 2.2.5 The mapping \( w \mapsto {w}^{-1} \) is an automorphism of Bruhat order (i.e., \( u \leq w \Leftrightarrow {u}^{-1} \leq {w}^{-1} \) ). | Proof. The subword relation is unaffected by reversing all expressions. (Remark: The result is also easy to derive directly from Definition 2.1.1, see Exercise 1 ). \( ▱ \) | No |
Theorem 2.2.6 (Chain Property) If \( u < w \), there exists a chain \( u = \) \( {x}_{0} < {x}_{1} < \cdots < {x}_{k} = w \) such that \( \ell \left( {x}_{i}\right) = \ell \left( u\right) + i \), for \( 1 \leq i \leq k \) . | Proof. This follows directly from Lemma 2.2.1 and the Subword Property. \( ▱ \) | No |
Proposition 2.2.7 (Lifting Property) Suppose \( u < w \) and \( s \in {D}_{L}\left( w\right) \smallsetminus \) \( {D}_{L}\left( u\right) \) . Then, \( u \leq {sw} \) and \( {su} \leq w \) . | Proof. Let \( \alpha \prec \beta \) here denote the subword relation between a word \( \beta \) and a subword \( \alpha \) . Choose a reduced decomposition \( {sw} = {s}_{1}{s}_{2}\ldots {s}_{q} \) . Then, \( w = s{s}_{1}{s}_{2}\ldots {s}_{q} \) is also reduced, and there exists a reduced subword\n\n\[ u = {s}_{{i}_{1}... | Yes |
Proposition 2.2.9 Bruhat order is a directed poset. | Proof. We will use induction on \( \ell \left( u\right) + \ell \left( w\right) \), the \( \ell \left( u\right) + \ell \left( w\right) = 0 \) case being trivially correct. Choose \( s \in S \) so that \( {su} < u \) (we may assume that \( \ell \left( u\right) > 0) \) . By induction, there exists \( x \in W \) such that ... | Yes |
Lemma 2.2.10 Suppose that \( x < {xt} \) and \( y < {ty} \), for \( x, y \in W, t \in T \) . Then, \( {xy} < {xty} \) . | Proof. Suppose to the contrary that \( {xy} > {xty} = {t}^{\prime }{xy} \), where \( {t}^{\prime } = {xt}{x}^{-1} \) . Let \( x = {s}_{1}\ldots {s}_{k} \) and \( y = {s}_{1}^{\prime }\ldots {s}_{q}^{\prime } \) be reduced expressions. Then, by the Strong Exchange Property,\n\n\[ \n{t}^{\prime }{xy} = \left\{ \begin{arr... | Yes |
Proposition 2.3.2 The top element \( {w}_{0} \) of a finite group has the following properties:\n\n(i) \( {w}_{0}^{2} = e \).\n\n(ii) \( \ell \left( {w{w}_{0}}\right) = \ell \left( {w}_{0}\right) - \ell \left( w\right) \), for all \( w \in W \).\n\n(iii) \( {T}_{L}\left( {w{w}_{0}}\right) = T \smallsetminus {T}_{L}\lef... | Proof. (i) Since \( \ell \left( {w}_{0}^{-1}\right) = \ell \left( {w}_{0}\right) \), uniqueness of \( {w}_{0} \) implies that \( {w}_{0}^{-1} = {w}_{0} \).\n\n(ii) The inequality \( \geq \) follows from \( \ell \left( {w}^{-1}\right) + \ell \left( {w{w}_{0}}\right) \geq \ell \left( {w}_{0}\right) \). For the opposite i... | Yes |
Corollary 2.3.3 (i) \( \ell \left( {{w}_{0}w}\right) = \ell \left( {w}_{0}\right) - \ell \left( w\right) \), for all \( w \in W \) . | Proof. \( \ell \left( {{w}_{0}w}\right) = \ell \left( {{w}^{-1}{w}_{0}}\right) = \ell \left( {w}_{0}\right) - \ell \left( {w}^{-1}\right) = \ell \left( {w}_{0}\right) - \ell \left( w\right) \). | Yes |
Proposition 2.3.4 For Bruhat order on a finite Coxeter group, the following hold:\n\n(i) \( w \mapsto w{w}_{0} \) and \( w \mapsto {w}_{0}w \) are antiautomorphisms.\n\n(ii) \( w \mapsto {w}_{0}w{w}_{0} \) is an automorphism. | The top element \( {w}_{0} \) in the symmetric group \( {S}_{n} \) is the \ | No |
Corollary 2.3.6 If \( \left( {W, S}\right) \) is irreducible and \( \left| S\right| \geq 3 \), then the automorphism group of Bruhat order is generated by the diagram automorphisms and the mapping \( x \mapsto {x}^{-1} \) . | For instance, the diagram of type \( {A}_{n}, n \geq 2 \), has a unique nontrivial automorphism, and this, in fact, induces the mapping \( x \mapsto {w}_{0}x{w}_{0} \) . Hence, the automorphism group of Bruhat order of the symmetric group \( {S}_{n}, n \geq 4 \) , is the dihedral group of order 4 generated by \( x \map... | Yes |
Proposition 2.4.1 (i) \( \left( {{W}_{J}, J}\right) \) is a Coxeter group. | Proof. Let \( w \in {W}_{J} \) . By definition, \( w = {s}_{1}{s}_{2}\ldots {s}_{q} \), for some \( {s}_{i} \in J \), and by the Deletion Condition, we may assume that this is reduced in \( W \), and hence in \( {W}_{J} \) . This proves (ii). Since \( {\ell }_{J}\left( w\right) = \ell \left( w\right) \), the Exchange P... | No |
Lemma 2.4.3 An element \( w \) belongs to \( {W}^{J} \) if and only if no reduced expression for \( w \) ends with a letter from \( J \) . | Proof. This follows from Corollary 1.4.6. \( ▱ \) | No |
Proposition 2.4.4 Let \( J \subseteq S \) . Then, the following hold:\n\n(i) Every \( w \in W \) has a unique factorization \( w = {w}^{J} \cdot {w}_{J} \) such that \( {w}^{J} \in {W}^{J} \) and \( {w}_{J} \in {W}_{J} \) .\n\n(ii) For this factorization, \( \ell \left( w\right) = \ell \left( {w}^{J}\right) + \ell \lef... | Proof. (Existence) Choose \( {s}_{1} \in J \) so that \( w{s}_{1} < w \), if such \( {s}_{1} \) exists. Continue choosing \( {s}_{i} \in J \) so that \( w{s}_{1}\ldots {s}_{i} < w{s}_{1}\ldots {s}_{i - 1} \) as long as such \( {s}_{i} \) can be found. The process must end after at most \( \ell \left( w\right) \) steps.... | Yes |
Proposition 2.4.8 For \( x, y \in {S}_{n}^{\left( k\right) } \), the following are equivalent:\n\n(i) \( x \leq y \) .\n\n(ii) \( {x}_{i} \leq {y}_{i} \), for \( 1 \leq i \leq k \) .\n\n(iii) \( {x}_{i} \geq {y}_{i} \), for \( k + 1 \leq i \leq n \) . | Proof. (i) \( \Rightarrow \) (ii). This is an immediate consequence of Theorem 2.1.5.\n\n(ii) \( \Rightarrow \) (i). Suppose that \( {x}_{j} < {y}_{j} \) for some \( 1 \leq j \leq k \) and \( {x}_{i} = {y}_{i} \) for all \( j + 1 \leq i \leq k \) . Then, \( {x}_{j} + 1 = {x}_{m} \) for some \( m > k \) (since \( {x}_{j... | No |
Proposition 2.5.1 The map \( {P}^{J} \) is order-preserving. | Proof. Suppose that \( {w}_{1} \leq {w}_{2} \) in \( W \) . We will show that \( {w}_{1}^{J} \leq {w}_{2}^{J} \) by induction on \( \ell \left( {w}_{2}\right) \) .\n\nTo begin with, note that \( {w}_{1}^{J} \leq {w}_{1} \leq {w}_{2} \) . Hence, if \( {w}_{2}^{J} = {w}_{2} \), we are done. If not, then there exists some... | Yes |
Corollary 2.5.2 Suppose \( u \in {W}^{J}, w \in W \) and \( u \vartriangleleft w \) . Then, either \( w = \) us, for some \( s \in J \), or \( w \in {W}^{J} \) . | Proof. If \( w \notin {W}^{J} \), then \( u \leq {P}^{J}\left( w\right) < w \) . | No |
Corollary 2.5.3 \( {W}^{J} \) is a directed poset. | Proof. This follows from Propositions 2.2.9 and 2.5.1. \( ▱ \) | Yes |
Proposition 2.5.4 Let \( \\left( {W, S}\\right) \) be finite, \( J \\subseteq S \) . Then,\n\n\[ \n\\alpha : x \\mapsto {w}_{0}x{w}_{0}\\left( J\\right)\n\]\n\ndefines an antiautomorphism \( \\alpha : {W}^{J} \\rightarrow {W}^{J} \) of Bruhat order (that is, \( x \\leq y \\Leftrightarrow \\alpha \\left( x\\right) \\geq... | Proof. We have that \( x \\in {W}^{J} \\Rightarrow x{w}_{0}\\left( J\\right) \\in {\\mathcal{D}}_{J}^{S} \\Rightarrow {w}_{0}x{w}_{0}\\left( J\\right) \\in {W}^{J} \), by Proposition 2.4.4 and Corollary 2.3.3. Furthermore, if \( x, y \\in {W}^{J} \), then \( x \\leq y \\Rightarrow x{w}_{0}\\left( J\\right) \\leq y{w}_{... | Yes |
Theorem 2.5.5 (Chain Property) If \( u < w \) in \( {W}^{J} \), then there exist elements \( {w}_{i} \in {W}^{J},\ell \left( {w}_{i}\right) = \ell \left( u\right) + i \), for \( 0 \leq i \leq k \), such that \( u = {w}_{0} < \) \( {w}_{1} < \cdots < {w}_{k} = w. \) | Proof. It suffices to construct \( {w}_{1} \) ; the rest follows via induction. Let \( w = \) \( {s}_{1}{s}_{2}\ldots {s}_{q} \), and take a reduced subword expression\n\n\[ u = {s}_{1}\ldots {\widehat{s}}_{{i}_{1}}\ldots {\widehat{s}}_{{i}_{k}}\ldots {s}_{q},\;1 \leq {i}_{1} < \cdots < {i}_{k} \leq q, \]\n\n such that... | Yes |
Theorem 2.6.3 (Tableau Criterion) For \( x, y \in {S}_{n} \), let \( {x}_{i, k} \) be the \( i \) - th element in the increasing rearrangement of \( {x}_{1},{x}_{2},\ldots ,{x}_{k} \), and similarly define \( {y}_{i, k} \) . Then, the following are equivalent:\n\n(i) \( x \leq y \) .\n\n(ii) \( {x}_{i, k} \leq {y}_{i, ... | Proof. Condition (ii) can, as shown by Proposition 2.4.8, be restated as saying that \( {P}^{S\smallsetminus \{ k\} }\left( x\right) \leq {P}^{S\smallsetminus \{ k\} }\left( y\right) \) for all \( k \in {D}_{R}\left( x\right) \) . Similarly, condition (iii) says that \( {P}^{S\smallsetminus \{ k\} }\left( {{w}_{0}y}\ri... | Yes |
Example 2.7.1 Let \( W \) be the dihedral group of order 6 on two generators \( S = \{ a, b\} \) (or equivalently, the symmetric group \( {S}_{3} \) ). Its Bruhat order is depicted in Figure 2.9. | Choosing \ | No |
Lemma 2.7.2 There is at most one chain \( \mathbf{m} \in \mathcal{M}\left( {u, v}\right) \) for which \( \lambda \left( \mathbf{m}\right) \) is increasing (meaning that \( {\lambda }_{1}\left( \mathbf{m}\right) < {\lambda }_{2}\left( \mathbf{m}\right) < \cdots < {\lambda }_{k}\left( \mathbf{m}\right) \) ). | Proof. The statement is clear for intervals of length 1 , so we may inductively suppose that it has been shown for length \( k - 1 \) . Suppose that there are two maximal chains \( \mathbf{m} : w = {x}_{0} \vartriangleright {x}_{1} \vartriangleright \cdots \vartriangleright {x}_{k} = u \) and \( {\mathbf{m}}^{\prime } ... | Yes |
Lemma 2.7.4 (i) There is a unique chain \( {\mathbf{m}}_{0} \in {\mathcal{M}}^{J}\left( {u, w}\right) \) such that \( \lambda \left( {\mathbf{m}}_{0}\right) \) is increasing. | Proof. A chain \( \mathbf{m} \) with increasing \( \lambda \left( \mathbf{m}\right) \) is constructed in Theorem 2.5.5, and it is unique by Lemma 2.7.2. This proves part (i). | Yes |
Theorem 2.7.7 The order complex of \( {\left( u, w\right) }^{J} \) is PL homeomorphic to\n\n(i) the sphere \( {\mathbb{S}}^{\ell \left( {u, w}\right) - 2} \), if \( {\left( u, w\right) }^{J} \) is full;\n\n(ii) the ball \( {\mathbb{B}}^{\ell \left( {u, w}\right) - 2} \), otherwise. | Proof. The complex \( \Delta \left( {\left( u, w\right) }^{J}\right) \) is pure \( \left( {\ell \left( {u, w}\right) - 2}\right) \) -dimensional, and Lemma 2.7.3 implies that it is thin if \( {\left( u, w\right) }^{J} \) is full and subthin otherwise. Hence, Theorem 2.7.5 and Fact A2.4.3 force the conclusion. \( ▱ \) | Yes |
Corollary 2.7.8 Suppose that \( \ell \left( {u, w}\right) = 3 \) . Then, the closed interval \( \left\lbrack {u, w}\right\rbrack \) is a \( k \) -crown, for some \( k \geq 2 \) . | Proof. The order complex of \( \left( {u, w}\right) \) triangulates the circle \( {\mathbb{S}}^{1} \) . The \( k \) -crown clearly has the only possible isomorphism type. \( ▱ \) | No |
Example 2.7.9 Let \( \\left( {W, S}\\right) \) be given by the Coxeter diagram\n\n\n\nLet \( u = {\\left( abc\\right) }^{n - 1} \) and \( w = {\\left( abc\\right) }^{n} \) . Then, \( u < w,\\ell \\left( {u, w}\\right) ... | To see this, observe that each of the \( {3n} \) letters in the reduced word \( w = {abcabc}\\ldots {abc} \) can be deleted, creating a reduced subword of length \( {3n} - 1 \), which uniquely represents an element of \( \\left\\lbrack {u, w}\\right\\rbrack \) . | Yes |
Corollary 2.7.10 \( {\mu }^{J}\left( {u, w}\right) = \left\{ \begin{array}{ll} {\left( -1\right) }^{\ell \left( {u, w}\right) }, & \text{ if }{\left\lbrack u, w\right\rbrack }^{J}\text{ is full,} \\ 0, & \text{ otherwise. } \end{array}\right. \) | Computing on the full Bruhat order of \( W \), the corollary specializes to say that \( \mu \left( {u, w}\right) = {\left( -1\right) }^{\ell \left( {u, w}\right) } \) for all \( u \leq w \) . From definition (A2.1), one sees that this is equivalent to the following: | No |
Corollary 2.7.11 In a closed interval \( \left\lbrack {u, w}\right\rbrack, u < w \), the number of elements of odd length equals the number of elements of even length. | The last two corollaries can, of course, be given direct combinatorial proofs. If there exists some \( s \in {D}_{L}\left( w\right) \smallsetminus {D}_{L}\left( u\right) \), then the mapping \( x \mapsto \) \( {sx} \) matches the odd-length and the even-length elements of \( \left\lbrack {u, w}\right\rbrack \), as is e... | No |
Theorem 2.7.12 Suppose that \( \ell \left( {u, w}\right) \geq 2 \) . Then, there exists a regular \( {CW} \) complex \( {\Gamma }_{u, w} \), uniquely determined up to cellular homeomorphism, whose cell poset is isomorphic to \( \left( {u, w}\right) \) and such that \( \begin{Vmatrix}{\Gamma }_{u, w}\end{Vmatrix} \cong ... | Proof. Let \( X = \parallel \Delta \left( \left( {u, w}\right) \right) \parallel \), the geometric realization of the order complex of \( \left( {u, w}\right) \) . Then, by Theorem 2.7.7, \( X \cong {\mathbb{S}}^{\ell \left( {u, w}\right) - 2} \) . For each \( z \in \left( {u, w}\right) \) , let \( {\sigma }_{z} = \par... | Yes |
Corollary 2.7.14 Suppose that \( \ell \left( {u, w}\right) \geq 2 \) . Then, there exists a signature \( \operatorname{sg} : \operatorname{Cov}\left\lbrack {u, w}\right\rbrack \rightarrow \{ + 1, - 1\} \) such that the following hold:\n\n(i) \( \operatorname{sg} \) is balanced.\n\n(ii) \( \operatorname{Im}{d}_{i} = \op... | Proof. Extend the cell complex \( {\Gamma }_{u, w} \) by attaching a \( \left( {\ell \left( {u, w}\right) - 1}\right) \) -cell via some homeomorphism of its boundary onto \( \begin{Vmatrix}{\Gamma }_{u, w}\end{Vmatrix} \) . This gives a regular CW complex \( {\widehat{\Gamma }}_{u, w} \) which decomposes the ball \( {\... | Yes |
Proposition 3.1.3 \( u{ \leq }_{R}w \Leftrightarrow {T}_{L}\left( u\right) \subseteq {T}_{L}\left( w\right) \) . | Proof. If \( u = {s}_{1}{s}_{2}\ldots {s}_{k} \) and \( w = {s}_{1}{s}_{2}\ldots {s}_{k}{s}_{k + 1}\ldots {s}_{q} \) are reduced, then\n\n\[ \n{T}_{L}\left( u\right) = \left\{ {{s}_{1}{s}_{2}\ldots {s}_{i}\ldots {s}_{2}{s}_{1} : 1 \leq i \leq k}\right\} \n\]\n\n\[ \n\subseteq \left\{ {{s}_{1}{s}_{2}\ldots {s}_{i}\ldots... | Yes |
Proposition 3.1.5 For weak order on a finite \( W \), the following hold:\n\n(i) \( w \mapsto {w}_{0}w \) and \( w \mapsto w{w}_{0} \) are antiautomorphisms.\n\n(ii) \( w \mapsto {w}_{0}w{w}_{0} \) is an automorphism. | Proof. This follows from the length formulas in Proposition 2.3.2 and its corollary and from the fact that if \( s \in S \), then \( s{w}_{0} = {w}_{0}{s}^{\prime } \) for some \( {s}^{\prime } \in S \) (a consequence of \( {w}_{0}S{w}_{0} = S \) ). For instance, if \( w{ \vartriangleleft }_{R}{ws} \), then \( {w}_{0}w... | Yes |
Proposition 3.1.6 If \( u{ \leq }_{R}w \), then \( {\left\lbrack u, w\right\rbrack }_{R} \cong {\left\lbrack e,{u}^{-1}w\right\rbrack }_{R} \) . | Proof. We will show that the mapping \( x \mapsto {ux} \) is a poset isomorphism \( {\left\lbrack e,{u}^{-1}w\right\rbrack }_{R} \rightarrow {\left\lbrack u, w\right\rbrack }_{R} \) . The basic properties of the length function give:\n\n\[ \ell \left( w\right) = \ell \left( u\right) + \ell \left( {{u}^{-1}w}\right) \]\... | Yes |
Corollary 3.2.2 Right order on a finite Coxeter group \( \left( {W, S}\right) \) together with the translation \( x \mapsto x{w}_{0} \) gives \( W \) the structure of an ortholattice. | Proof. This follows from Propositions 2.3.2(iii) and 3.1.3. | No |
Lemma 3.2.3 Let \( J \subseteq S \) . Then, \( \bigvee J \) exists if and only if \( {W}_{J} \) is finite, and if so, \( \bigvee J = {w}_{0}\left( J\right) \) . In particular, if \( \left| {W}_{J}\right| = \infty \), then \( J \) has no upper bound. | Proof. If \( {W}_{J} \) is finite, then \( {D}_{L}\left( {{w}_{0}\left( J\right) }\right) = J \) shows, keeping Corollary 1.4.6 in mind, that \( {w}_{0}\left( J\right) \) is an upper bound to the set \( J \) . We show that, conversely, if \( J \) has an upper bound \( w \), or equivalently if \( J \subseteq {D}_{L}\lef... | Yes |
Lemma 3.2.4 Let \( w \in W, J \subseteq S \) with \( {W}_{J} \) finite.\n\n(i) If \( w{ \vartriangleleft }_{R}{ws} \) for all \( s \in J \), then \( \bigvee \{ {ws} : s \in J\} = w{w}_{0}\left( J\right) \).\n\n(ii) If \( {ws}{ \vartriangleleft }_{R}w \) for all \( s \in J \), then \( \bigwedge \{ {ws} : s \in J\} = w{w... | Proof. The conditions mean that \( w \in {W}^{J} \) (minimal coset representatives) and \( w \in {\mathcal{D}}_{J}^{S} \) (maximal coset representatives), respectively. Part (i) then follows easily via Propositions 3.1.6 and 3.2.3. Part (ii) requires a few more steps that we leave to the reader. \( ▱ \) | No |
Theorem 3.2.7 Suppose that \( u{ < }_{R}w \) and \( \ell \left( {u, w}\right) \geq 2 \) . Then, the order complex of the open interval \( {\left( u, w\right) }_{R} \) is as follows:\n\n(i) Homotopy equivalent to the sphere \( {\mathbb{S}}^{\left| J\right| - 2} \), if \( w = u{w}_{0}\left( J\right) \) for some \( J \sub... | Proof. By Proposition 3.1.6 we may assume that \( u = e \) . For each \( x \in {\left( e, w\right) }_{R} \), let \( f\left( x\right) = \bigvee \left\{ {s \in S : s{ \leq }_{R}x}\right\} \) . Lemma 3.2.3 shows that, equivalently, \( f\left( x\right) = {w}_{0}\left( {{D}_{L}\left( x\right) }\right) \) . Then, \( f \) is ... | Yes |
Corollary 3.2.8\n\n\[ \n{\mu }_{R}\left( {u, w}\right) = \left\{ \begin{array}{ll} {\left( -1\right) }^{\left| J\right| } & ,\text{ if }w = u{w}_{0}\left( J\right) \text{ for some }J \subseteq S; \\ 0 & ,\text{ otherwise. } \end{array}\right. \n\] | Proof. This follows since the Möbius function is the reduced Euler characteristic of the order complex; see Fact A2.3.1. \( ▱ \) | No |
Theorem 3.3.1 (Word Property) Let \( \\left( {W, S}\\right) \) be a Coxeter group and \( w \\in W \) .\n\n(i) Any expression \( {s}_{1}{s}_{2}\\ldots {s}_{q} \) for \( w \) can be transformed into a reduced expression for \( w \) by a sequence of nil-moves and braid-moves.\n\n(ii) Every two reduced expressions for \( w... | Proof. We begin by proving (ii). This will be done by induction on \( \\ell \\left( w\\right) \) , the result being clear if \( \\ell \\left( w\\right) \\leq 1 \) . Assume that \( \\ell \\left( w\\right) > 1 \) and let\n\n\[ w = s{s}_{2}\\ldots {s}_{k} = {s}^{\\prime }{s}_{2}^{\\prime }\\ldots {s}_{k}^{\\prime }\n\]\n\... | Yes |
Lemma 3.4.1 Let \( w \in W \) . Then, \( w \) is distinguished if and only if \( w \in \) \( {}^{\left\lbrack n - 1\right\rbrack }W \) . | Proof. We know from Proposition 2.4.3 that \( w \in {}^{\left\lbrack n - 1\right\rbrack }W \) if and only if no reduced word for \( w \) begins with one of the letters \( {s}_{1},\ldots ,{s}_{n - 1} \) . However, this is precisely what it means for \( w \) to be distinguished. | Yes |
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