Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Proposition 4.28. Let \( M \) be a finite-dimensional A-module. The following assertions are equivalent.\n\n(i) \( M \) is semisimple.\n\n(ii) \( M \) is completely reducible.\n\n(iii) \( M = \mathop{\sum }\limits_{{i \in I}}{M}_{i} \) is a sum of simple submodules \( {M}_{i} \). | Proof. We first prove the implication (ii) \( \Rightarrow \) (iii). Assume that \( M \) is nonzero and completely reducible. Since \( M \) is finite-dimensional over \( K \), it must have nonzero submodules of minimal dimension as vector spaces over \( K \) ; such submodules are necessarily simple. Consider the sum \( ... | Yes |
Proposition 4.29. Let \( M \) be a finite-dimensional semisimple A-module. Any A-submodule and any quotient A-module of \( M \) is semisimple. | Proof. Let \( {M}_{0} \) be a submodule of \( M \) . Let \( {M}_{0}^{\prime } \) be the sum of all simple submodules of \( {M}_{0} \) . Since by Proposition 4.28, \( M \) is completely reducible, \( M = {M}_{0}^{\prime } \oplus {M}^{\prime \prime } \) for some submodule \( {M}^{\prime \prime } \) of \( M \) . Together ... | Yes |
Proposition 4.30. (a) Let \( M \) and \( {M}^{\prime } \) be simple \( A \) -modules. If \( M \) and \( {M}^{\prime } \) are not isomorphic as \( A \) -modules, then \( {\operatorname{Hom}}_{A}\left( {M,{M}^{\prime }}\right) = 0 \) . | Proof. (a) Let \( f \in {\operatorname{Hom}}_{A}\left( {M,{M}^{\prime }}\right) \) . The kernel \( \operatorname{Ker}\left( f\right) \) of \( f \) is a submodule of \( M \) . Since \( M \) is simple, \( \operatorname{Ker}\left( f\right) = M \) or \( \operatorname{Ker}\left( f\right) = 0 \) . In the first case, \( f = 0... | Yes |
Proposition 4.32. If for some families \( \{ d\left( \lambda \right) {\} }_{\lambda \in \Lambda },\{ e\left( \lambda \right) {\} }_{\lambda \in \Lambda } \) of nonnegative integers, there is an A-module isomorphism\n\n\[ \n{\bigoplus }_{\lambda \in \Lambda }{V}_{\lambda }^{d\left( \lambda \right) } \cong {\bigoplus }_{... | Proof. Pick \( {\lambda }_{0} \in \Lambda \) and set \( D = {\operatorname{End}}_{A}\left( {V}_{{\lambda }_{0}}\right) \) . By Proposition 4.30 (a),\n\n\[ \n\operatorname{Hom}\left( {{\bigoplus }_{\lambda \in \Lambda }{V}_{\lambda }^{d\left( \lambda \right) },{V}_{{\lambda }_{0}}}\right) \cong \mathop{\prod }\limits_{{... | Yes |
Proposition 4.34. Let \( V \) be a finite-dimensional left vector space over a division ring \( D \) . Then the algebra \( {\operatorname{End}}_{D}\left( V\right) \) is simple. | Proof. Pick a basis \( \left\{ {{v}_{1},\ldots ,{v}_{d}}\right\} \) of \( V \) . We have \( V = D{v}_{1} \oplus \cdots \oplus D{v}_{d} \) . For \( i, j \in \{ 1,\ldots, d\} \), define \( {f}_{i, j} \in A = {\operatorname{End}}_{D}\left( V\right) \) by \( {f}_{i, j}\left( {v}_{k}\right) = {\delta }_{j, k}{v}_{i} \) for ... | Yes |
Proposition 4.35. For any simple algebra \( A \), there is a division ring \( D \) and a finite-dimensional \( D \) -vector space \( V \) such that \( A \cong {\operatorname{End}}_{D}\left( V\right) \) . | Proof. Pick a left ideal \( V \subset A \) of \( A \) of minimal positive dimension over \( K \) (possibly, \( V = A \) ). The ideal \( V \) is an \( A \) -module, and by the minimality condition, it is a simple module. By Proposition \( {4.30}\left( \mathrm{\;b}\right), D = {\operatorname{End}}_{A}\left( V\right) \) i... | Yes |
Proposition 4.37. Let \( A \) be a simple algebra. Any nonzero left ideal \( I \) of \( A \) of minimal dimension is a simple \( A \) -module, and any nonzero simple \( A \) -module is isomorphic to \( I \) . | Proof. Let \( I \) be a nonzero left ideal of \( A \) of minimal dimension. Any \( A \) -submodule \( {I}^{\prime } \) of \( I \) is a left ideal of \( A \) . By the minimality hypothesis on \( I \), we must have \( {I}^{\prime } = 0 \) or \( {I}^{\prime } = I \) . Therefore, \( I \) is a simple \( A \) -module; it is ... | Yes |
Corollary 4.38. Every simple algebra has a nonzero simple module. It is finite-dimensional and unique up to isomorphism. | Proof. Every finite-dimensional algebra has a nonzero left ideal of minimal dimension. Therefore both claims follow directly from Proposition 4.37. | No |
Proposition 4.39. Any finite-dimensional module over a simple algebra is semisimple. | Proof. Let \( A \) be a simple algebra. Consider \( A \) as a (left) module over itself. Let us first prove that this \( A \) -module is semisimple.\n\nBy Proposition 4.35, we may assume that \( A = {\operatorname{End}}_{D}\left( V\right) \) for some division ring \( D \) and some finite-dimensional \( D \) -vector spa... | Yes |
Corollary 4.40. With the notation above, \( A \cong {\operatorname{End}}_{D}\left( M\right) \) and\n\n\[{\dim }_{K}A = \frac{{\left( {\dim }_{K}M\right) }^{2}}{{\dim }_{K}D}.\] | Proof. The division ring \( D = {\operatorname{End}}_{A}\left( M\right) \) acts on \( M \), turning \( M \) into a left \( D \) -vector space of finite dimension over \( K \subset D \) . Such a vector space has a finite basis over \( D \) of cardinality, say \( d \) . Lemma 4.36 and Proposition 4.37 imply that \( A \co... | Yes |
Lemma 4.41. For all \( a, b, c \in A \) ,\n\n\[ \langle a, b\rangle = \langle b, a\rangle \;\text{ and }\;\langle {ab}, c\rangle = \langle {ab}, c\rangle = \langle b,{ca}\rangle . \] | Proof. The equality \( \langle {ab}, c\rangle = \langle a,{bc}\rangle \) follows from the formula \( {\mathrm{R}}_{\left( {ab}\right) c} = {\mathrm{R}}_{a\left( {bc}\right) } \) . The proof of the equality \( \langle a, b\rangle = \langle b, a\rangle \) relies on a well-known property of the trace, namely \( \operatorn... | Yes |
Lemma 4.42. The radical \( J\\left( A\\right) \) is a two-sided ideal of \( A \) . | Proof. Let \( a \\in A \) and \( b \\in J\\left( A\\right) \) . We have to check that \( {ab},{ba} \\in J\\left( A\\right) \) . Using Lemma 4.41, for all \( c \\in A \) ,\n\n\[ \n\\langle {ab}, c\\rangle = \\langle b,{ca}\\rangle = 0\\;\\text{ and }\\;\\langle {ba}, c\\rangle = \\langle b,{ac}\\rangle = 0.\n\] | Yes |
Proposition 4.43. If \( A \) is the product of a finite family \( {\left\{ {A}_{\lambda }\right\} }_{\lambda \in \Lambda } \) of finite-dimensional algebras, then\n\n\[ J\left( A\right) = \mathop{\prod }\limits_{{\lambda \in \Lambda }}J\left( {A}_{\lambda }\right) \] | Proof. Under the assumptions, \( A \) can be identified with the direct sum \( {\bigoplus }_{\lambda \in \Lambda }{A}_{\lambda } \) . It follows from the definition of the product in \( A \) that each right multiplication \( {R}_{a} \in {\operatorname{End}}_{K}\left( A\right) \), where \( a = {\left( {a}_{\lambda }\rig... | Yes |
Proposition 4.44. Any nilpotent left ideal of a finite-dimensional algebra \( A \) is contained in \( J\left( A\right) \) . | Proof. Let \( I \) be a nilpotent left ideal of \( A \) . To prove that \( I \subset J\left( A\right) \), we have to check that \( \langle a, b\rangle = 0 \) for all \( a \in I \) and \( b \in A \) . Set \( c = {ba} \in I \) . The ideal \( I \) being nilpotent, \( {c}^{N} = 0 \) for some \( N \geq 1 \) . Hence, \( {\le... | Yes |
Proposition 4.46. A finite-dimensional algebra \( A \) is semisimple if and only if for some basis \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) of \( A \) , | Proof. The nondegeneracy of a symmetric bilinear form \( \langle \) , \( \rangle {onafinite} - \) dimensional vector space with basis \( \left\{ {{a}_{1},\ldots ,{a}_{n}}\right\} \) is equivalent to the nonvanishing of the determinant \( \det \left( {\left\langle {a}_{i},{a}_{j}\right\rangle }_{i, j = 1,\ldots, n}\righ... | No |
Lemma 4.48. Let \( A \) be an algebra and \( {\left\{ {A}_{\lambda }\right\} }_{\lambda \in \Lambda } \) a finite family of subalgebras of \( A \) such that \( A = {\bigoplus }_{\lambda \in \Lambda }{A}_{\lambda } \) and \( {A}_{\lambda }{A}_{\mu } = 0 \) for any distinct \( \lambda ,\mu \in \Lambda \) . Then the follo... | Proof. (a) We have\n\n\[ A{A}_{\mu } = {\bigoplus }_{\lambda \in \Lambda }{A}_{\lambda }{A}_{\mu } = {A}_{\mu }{A}_{\mu } \subset {A}_{\mu } \]\n\nThe inclusion \( {A}_{\mu }A \subset {A}_{\mu } \) is proved in a similar way.\n\n(b) If \( a \in A \), then\n\n\[ \mathop{\sum }\limits_{{\lambda \in \Lambda }}a{1}_{\lambd... | Yes |
Corollary 4.51. Let \( J \) be a two-sided ideal of a semisimple algebra \( A \). Then \( J \) and the quotient algebra \( A/J \) are semisimple algebras. | Proof. Consider the splitting \( A = {\bigoplus }_{\lambda \in \Lambda }{A}_{\lambda } \) of \( A \) as in Theorem 4.49. By Example 4.47 (ii), each \( {A}_{\lambda } \) is semisimple. We have seen in the proof of Theorem 4.49 that there is a set \( {\Lambda }_{0} \subset \Lambda \) such that \( J = {\bigoplus }_{\lambd... | Yes |
Proposition 4.52. Let \( A \) be a semisimple algebra and \( {\left\{ {A}_{\lambda }\right\} }_{\lambda \in \Lambda } \) the family of simple subalgebras of \( A \) provided by Theorem 4.49. For any nonzero simple \( A \) -module \( M \), there is a unique \( \lambda \in \Lambda \) such that \( M = {A}_{\lambda }M \) .... | Proof. Let \( M \) be a nonzero simple \( A \) -module. Each \( {A}_{\lambda }M \) is an \( A \) -submodule of \( M \) . We can write \( M \) as a sum of these submodules:\n\n\[ M = {AM} = \mathop{\sum }\limits_{{\lambda \in \Lambda }}{A}_{\lambda }M \]\n\n(4.29)\n\nSince \( M \neq 0 \), there is \( \lambda \in \Lambda... | Yes |
Theorem 4.53. Any finite-dimensional module over a semisimple algebra is semisimple. | Proof. Consider a finite-dimensional module \( M \) over a semisimple algebra \( A \) . Expand \( A \) as a product of simple subalgebras \( {\left\{ {A}_{\lambda }\right\} }_{\lambda \in A} \) as in Theorem 4.49. Each vector space \( {A}_{\lambda }M \subset M \) is a finite-dimensional module over \( {A}_{\lambda } \)... | Yes |
Corollary 4.55. If the ground field \( K \) is algebraically closed, then the algebra homomorphism\n\n\[ A \rightarrow \mathop{\prod }\limits_{{\lambda \in \Lambda }}{\operatorname{End}}_{K}\left( {V}_{\lambda }\right) \]\n\nobtained as the product over \( \Lambda \) of the algebra homomorphisms \( A \rightarrow {\oper... | Proof. Applying Proposition 4.30 (c) to the simple \( A \) -module \( {V}_{\lambda } \), we obtain \( {\dim }_{K}{D}_{\lambda } = 1 \) . Thus, \( {D}_{\lambda } = K \) . The corollary is then a reformulation of Theorem 4.54. | Yes |
Proposition 4.56. There is an isomorphism of \( S \) -algebras\n\n\[ \varphi : S{ \otimes }_{R}{H}_{n}^{R}\left( q\right) \overset{ \cong }{ \rightarrow }{H}_{n}^{S}\left( \widetilde{q}\right) \]\n\nsuch that\n\n\[ {\left\langle \varphi \left( s \otimes x\right) ,\varphi \left( {s}^{\prime } \otimes {x}^{\prime }\right... | Proof. Set \( \varphi \left( {s \otimes {T}_{i}}\right) = s{T}_{i} \in {H}_{n}^{S}\left( \widetilde{q}\right) \) for \( s \in S \) and \( i = 1,\ldots, n - 1 \) . It is easy to check that this defines a homomorphism of \( S \) -algebras\n\n\[ \varphi : S{ \otimes }_{R}{H}_{n}^{R}\left( q\right) \rightarrow {H}_{n}^{S}\... | Yes |
Theorem 5.1. For all \( n \geq 1 \) ,\n\n\[ \mathop{\sum }\limits_{{\lambda \dashv n}}{\left( {f}^{\lambda }\right) }^{2} = n! \] | A proof of this theorem will be given in Section 5.2.4. | No |
Lemma 5.2. Let \( T \) be a tableau with \( n \) boxes and \( i, j \in \{ 1,\ldots, n - 1\} \) . (a) Then \[ {d}_{{s}_{j}T}\left( i\right) = \left\{ \begin{array}{ll} - {d}_{T}\left( i\right) & \text{ if }j = i, \\ {d}_{T}\left( i\right) & \text{ if }\left| {i - j}\right| \geq 2. \end{array}\right. \] | Proof. (a) Let \( \left( {r, s}\right) \) be the box of \( T \) with label \( i \) and \( \left( {{r}^{\prime },{s}^{\prime }}\right) \) the box of \( T \) with label \( i + 1 \) . Then \( \left( {r, s}\right) \) is the box of \( {s}_{i}T \) with label \( i + 1 \) and \( \left( {{r}^{\prime },{s}^{\prime }}\right) \) i... | Yes |
Lemma 5.3. Let \( T \) be a standard tableau with \( n \) boxes.\n\n(a) If the labels \( i \) and \( i + 1 \) of \( T \) sit in the same row, then \( {d}_{T}\left( i\right) = 1 \) .\n\n(b) If \( i \) and \( i + 1 \) sit in the same column, then \( {d}_{T}\left( i\right) = - 1 \) .\n\n(c) If \( i \) and \( i + 1 \) sit ... | Proof. Let \( \left( {r, s}\right) \) and \( \left( {{r}^{\prime },{s}^{\prime }}\right) \) be the boxes of \( T \) with labels \( i \) and \( i + 1 \), respectively.\n\n(a) If \( i \) and \( i + 1 \) sit in the same row, then they necessarily occupy adjacent boxes, so that \( {r}^{\prime } = r \) and \( {s}^{\prime } ... | Yes |
Lemma 5.4. Let \( \lambda ,{\lambda }^{\prime } \) be distinct partitions of the same positive integer. Then there is at most one partition \( \mu \) such that \( \mu \hookrightarrow \lambda \) and \( \mu \hookrightarrow {\lambda }^{\prime } \), and there is at most one partition \( \nu \) such that \( \lambda \hookrig... | Proof. Let \( \mu \) be such that \( \mu \hookrightarrow \lambda \) and \( \mu \hookrightarrow {\lambda }^{\prime } \) . Then \( D\left( \mu \right) \subset D\left( \lambda \right) \cap D\left( {\lambda }^{\prime }\right) \) and \( \operatorname{card}D\left( \mu \right) = n - 1 \), where \( n = \left| \lambda \right| =... | Yes |
Lemma 5.5. Let \( \lambda = \left( {{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{p}}\right) \) be an arbitrary partition. Suppose that there are \( \ell \) partitions \( \mu \) such that \( \mu \hookrightarrow \lambda \) . Then there are \( \ell + 1 \) partitions \( \nu \) such that \( \lambda \hookrightarrow \nu ... | Proof. It is clear that \( \left( {r, s}\right) \) is a corner of \( \lambda \) if and only if \( {\lambda }_{r} > {\lambda }_{r + 1} \) (here we identify \( \lambda \) with an infinite nonincreasing eventually zero sequence of integers). Thus, the number \( \ell \) of partitions \( \mu \) such that \( \mu \hookrightar... | Yes |
For any partition \( \lambda \), the number \( {f}^{\lambda } = \operatorname{card}{\mathcal{T}}_{\lambda } \) is equal to the number of oriented paths from \( \varnothing \) to \( \lambda \) in \( \mathcal{Y} \) . | The box with the largest label \( n \) in a standard tableau \( T \) of shape \( \lambda \dashv n \) is necessarily a corner. Removing this box, we obtain a partition \( {\lambda }^{\left( n - 1\right) } \) of \( n - 1 \) and a standard tableau of shape \( {\lambda }^{\left( n - 1\right) } \) . Removing the corner labe... | Yes |
For any partition \( \lambda \dashv n \geq 0 \) , \[ {D}^{n}\left( \lambda \right) = {f}^{\lambda }\varnothing \;\text{ and }\;{U}^{n}\left( \varnothing \right) = \mathop{\sum }\limits_{{\lambda \dashv n}}{f}^{\lambda }\lambda . \] | It follows from the definitions that for each \( k \geq 1 \) , \[ {D}^{k}\left( \lambda \right) = \mathop{\sum }\limits_{{{\lambda }^{\left( n - 1\right) } \hookrightarrow \lambda }}\mathop{\sum }\limits_{{{\lambda }^{\left( n - 2\right) } \hookrightarrow {\lambda }^{\left( n - 1\right) }}}\cdots \mathop{\sum }\limits_... | Yes |
Lemma 5.8 (The Heisenberg relation). We have \( {DU} - {UD} = \mathrm{{id}} \) . | Proof. Let \( \lambda \dashv n \geq 1 \) . By Lemma 5.4,\n\n\[ \n\left( {DU}\right) \left( \lambda \right) = \mathop{\sum }\limits_{{\lambda \hookrightarrow \nu }}D\left( \nu \right) = \mathop{\sum }\limits_{{\lambda \hookrightarrow \nu }}\left( {\mathop{\sum }\limits_{{{\lambda }^{\prime } \hookrightarrow \nu }}{\lamb... | Yes |
Lemma 5.10. If \( q \) is \( \left( {n - 1}\right) \) -regular, then\n\n\[ \n{a}_{T}\left( i\right) = q \Leftrightarrow {d}_{T}\left( i\right) = 1\;\text{ and }\;{a}_{T}\left( i\right) = - 1 \Leftrightarrow {d}_{T}\left( i\right) = - 1.\n\] | Proof. Set \( d = {d}_{T}\left( i\right) \) . Then\n\n\[ \n{a}_{T}\left( i\right) = q \Leftrightarrow {\left\lbrack d\right\rbrack }_{q} = {q}^{d - 1} \Leftrightarrow {\left\lbrack d - 1\right\rbrack }_{q} = 0.\n\]\n\nSince \( q \) is \( \left( {n - 1}\right) \) -regular and \( d < n \), the number \( {\left\lbrack d -... | Yes |
Theorem 5.11. Formula (5.12) defines the structure of a left \( {H}_{n}^{R}\left( q\right) \) -module on \( {V}_{\lambda } \) . | A proof of Theorem 5.11 will be given in Section 5.4. | No |
Proposition 5.13. For any partition \( \lambda \) of \( n \), there is a canonical isomorphism of \( {H}_{n - 1}^{R}\left( q\right) \) -modules\n\n\[{\left. {\left. {V}_{\lambda }\right| }_{{H}_{n - 1}^{R}\left( q\right) } = {\bigoplus }_{\mu \hookrightarrow \lambda }{V}_{\mu }\right| }_{\mu \hookrightarrow \lambda }\] | Proof. We observed in Section 5.1.5 that the label \( n \) in a standard tableau of shape \( \lambda \) sits necessarily in a corner of \( \lambda \) . Therefore we can partition the set of standard tableaux of shape \( \lambda \) according to the corner in which \( n \) sits. We thus obtain the partition\n\n\[{\mathca... | Yes |
Lemma 5.20. Let \( n \geq 3 \) and \( \alpha \in K \) . The only vector \( v \in {\mathcal{L}}_{n - 1} \) satisfying \( {V}_{i}v = {\alpha v} \) for all \( i = 1,\ldots, n - 1 \) is zero. | Proof. It is obvious from the form of \( {V}_{1} \) that its only eigenvalues are 1 and \( - t \) . Therefore, it suffices to establish the lemma for \( \alpha = 1 \) and \( \alpha = - t \) .\n\nIt is easy to check that the eigenspace of the action of \( {V}_{i} \) on \( {\mathcal{L}}_{n - 1} \) for the eigenvalue -1 i... | Yes |
Proposition 5.21. There is a unique structure of an \( {H}_{n}^{K}\left( t\right) \) -module on \( {\mathcal{L}}_{n - 1} \) such that each generator \( {T}_{i}\left( {i = 1,\ldots, n - 1}\right) \) acts on \( {\mathcal{L}}_{n - 1} \) by multiplication by the matrix \( - {V}_{i} \) . | Proof. We know from Section 3.3.1 that the matrices \( {V}_{1},\ldots ,{V}_{n - 1} \) satisfy relations (4.16) and (4.17). So do the matrices \( - {V}_{1},\ldots , - {V}_{n - 1} \) . It is easy to check that each \( {V}_{i} \) satisfies the equation\n\n\[ \left( {{V}_{i} - {I}_{n - 1}}\right) \left( {{V}_{i} + t{I}_{n ... | Yes |
Corollary 5.23. The reduced Burau representation \( {\psi }_{n}^{\mathrm{r}} : {B}_{n} \rightarrow {\mathrm{{GL}}}_{n - 1}\left( K\right) \) is irreducible. | Proof. The irreducibility of \( {\psi }_{n}^{\mathrm{r}} \) means that the only subspaces of \( {K}^{n - 1} \) preserved by \( {\psi }_{n}^{\mathrm{r}} \) are 0 and \( {K}^{n - 1} \) . If \( W \) is such a subspace, then\n\n\[ \left( {-{V}_{i}}\right) W = {V}_{i}W \subset W \]\n\nfor all \( i = 1,\ldots, n - 1 \) . By ... | Yes |
Lemma 5.25. Any nonempty word \( w = {e}_{{i}_{1}}\cdots {e}_{{i}_{r}} \) is equal in \( {A}_{n}\left( a\right) \) to a scalar multiple of a word in which the maximal generator appears exactly once. | Proof. We proceed by induction on the index \( p \) of \( w \) . If \( p = 1 \), then \( w \) is a positive power of \( {e}_{1} \) . From (5.22) we derive \( {e}_{1}^{i} = {a}^{i - 1}{e}_{1} \) for all \( i > 1 \) . Therefore, Lemma 5.25 holds for \( p = 1 \) .\n\nSuppose that Lemma 5.25 holds for all words of index \(... | Yes |
Theorem 5.29. Let \( q, a \in \mathbf{C} - \{ 0\} \) satisfy \( {a}^{2} = {\left( q + 1\right) }^{2}/q \) . (a) There is a surjective algebra homomorphism \( \Psi : {H}_{n}\left( q\right) \rightarrow {A}_{n}\left( a\right) \) such that \[ \Psi \left( {T}_{i}\right) = \frac{q + 1}{a}{e}_{i} - 1 \] for \( i = 1,\ldots, n... | Proof. (a) For \( i = 1,\ldots, n - 1 \), set \[ {t}_{i} = \Psi \left( {T}_{i}\right) = \frac{q + 1}{a}{e}_{i} - 1 \in {A}_{n}\left( a\right) . \] Formula (5.23) defines an algebra homomorphism \( \Psi : {H}_{n}\left( q\right) \rightarrow {A}_{n}\left( a\right) \), provided \( {t}_{1},\ldots ,{t}_{n - 1} \) satisfy rel... | Yes |
Lemma 5.30. If \( \lambda = \left( {{\lambda }_{1},{\lambda }_{2},\ldots ,{\lambda }_{p}}\right) \) is a partition of an integer \( n \geq 3 \) such that \( {\lambda }_{i} \in \{ 1,2\} \) for all \( i = 1,\ldots, p \), then \( {I}_{n}{V}_{\lambda } = 0 \) . | Proof. By Theorem 5.29 (c), it suffices to show that\n\n\[ X = 1 + {T}_{1} + {T}_{2} + {T}_{1}{T}_{2} + {T}_{2}{T}_{1} + {T}_{1}{T}_{2}{T}_{1} \in {H}_{3}\left( q\right) \subset {H}_{n}\left( q\right) \]\n\nacts trivially on \( {V}_{\lambda } \) . We proceed by induction on \( n \geq 3 \) .\n\nSuppose that \( n = 3 \) ... | Yes |
Corollary 5.32. Let \( n \geq 2 \) .\n\n(a) The dimension of \( {A}_{n}\left( a\right) \) as a complex vector space is given by\n\n\[ \n{\dim }_{\mathbf{C}}{A}_{n}\left( a\right) = \frac{1}{n + 1}\left( \begin{matrix} {2n} \\ n \end{matrix}\right) .\n\] | Proof. (a) For \( n \geq 3 \), this follows from Proposition 5.31, since\n\n\[ \n{\dim }_{\mathbf{C}}{A}_{n}\left( a\right) = {\dim }_{\mathbf{C}}{H}_{n}\left( q\right) - {\dim }_{\mathbf{C}}{I}_{n}.\n\]\n\nFor \( n = 2 \), the claim (a) is straightforward. | No |
Lemma 6.2. The relations \( \preccurlyeq \) and \( \succcurlyeq \) in a monoid are reflexive and transitive. | Proof. The reflexivity of \( \preccurlyeq \) follows from the identity \( a = {a1} \) ; and the transitivity, from the associativity of multiplication. The proofs for \( \succcurlyeq \) are similar. | Yes |
Lemma 6.3. If elements \( a, b \) of an atomic monoid \( M \) satisfy \( a \preccurlyeq b \) and \( b \preccurlyeq a \) , then \( a = b \) . Similarly, if \( a \succcurlyeq b \) and \( b \succcurlyeq a \), then \( a = b \) . | Proof. Since \( a \preccurlyeq b \preccurlyeq a \), there are \( u, v \in M \) such that \( b = {au} \) and \( a = {bv} \) . Then \( a = {auv} \) and\n\n\[ \parallel a\parallel = \parallel {auv}\parallel \geq \parallel a\parallel + \parallel u\parallel + \parallel v\parallel . \]\n\nThis implies that \( \parallel u\par... | Yes |
Lemma 6.4. If a monoid \( M \) has a weighted presentation \( \langle X \mid R\rangle \), then \( M \) is atomic and all its atoms are contained in the set \( X \) of generators. If \( M \) has a length-balanced presentation \( \langle X \mid R\rangle \), then the set of atoms of \( M \) coincides with \( X \) and \( \... | Proof. Let \( \ell : M \rightarrow \mathbf{N} \) be a monoid homomorphism such that \( \ell \left( x\right) \geq 1 \) for all generators \( x \in X \) . Then \( \ell \left( a\right) \geq 1 \) for all \( a \in M - \{ 1\} \) . If \( a \in M \) expands as a product \( {a}_{1}\cdots {a}_{r} \) with \( {a}_{1},\ldots ,{a}_{... | Yes |
Theorem 6.5. Let \( M \) be an atomic monoid and \( \sum \) a subset of \( M \) such that \( 1 \in \sum \) and the following three conditions hold:\n\n\( \left( { * }_{1}\right) \) All left divisors and all right divisors of elements of \( \sum \) belong to \( \sum \) .\n\n\( \left( { * }_{2}\right) \) For any \( a, b,... | Proof. The proof goes in five steps.\n\nStep 1. By \( \left( { * }_{3}\right) \), for any \( a, b \in \sum \), the set \( \{ x \in \sum \mid x \preccurlyeq b \) and \( {ax} \in \sum \} \) has a maximal element \( c \in \sum \) . Then \( b = {cd} \) for some \( d \in M \) . By \( \left( { * }_{1}\right), d \in \sum \) ,... | Yes |
Lemma 6.6. Under the assumptions of Theorem 6.5, the monoid \( {M}_{\sum } \) is left cancellative. | Proof. We need to show that \( {\xi \eta } = {\xi \theta } \Rightarrow \eta = \theta \) for \( \xi ,\eta ,\theta \in {M}_{\sum } \) . Suppose first that \( \xi = \left\lbrack a\right\rbrack \) for some \( a \in \sum \) . Then \( \alpha \left( {\xi \eta }\right) = {\alpha }_{2}\left( {a,\alpha \left( \eta \right) }\righ... | Yes |
Lemma 6.7. Let \( M,\sum \subset M \) satisfy the assumptions of Theorem 6.5. Given \( a, b \in {M}_{\sum } \), there is \( c \in {M}_{\sum } \) such that \( {ac} = {cb} \) if and only if there exist a sequence \( {a}_{0} = a,{a}_{1},\ldots ,{a}_{r} = b \) of elements of \( {M}_{\sum } \) and a sequence \( {c}_{1},\ldo... | Proof. If we have such sequences, then \( {ac} = {cb} \) for \( c = \left\lbrack {c}_{1}\right\rbrack \left\lbrack {c}_{2}\right\rbrack \cdots \left\lbrack {c}_{r}\right\rbrack \) . Conversely, let \( c \in {M}_{\sum } \) be such that \( {ac} = {cb} \) . We prove the assertion by induction on the length \( r \) of the ... | Yes |
Lemma 6.8. If \( \sum \) is a comprehensive subset of a monoid \( M \) such that all left divisors of elements of \( \sum \) belong to \( \sum \), then the monoid homomorphism \( p : {M}_{\sum } \rightarrow M \) is an isomorphism. | Proof. Since \( \sum \) contains a set of generators of \( M \), the homomorphism \( p \) is surjective. We need only to prove its injectivity. Observe first that if \( {a}_{1},{a}_{2},\ldots ,{a}_{n} \) are elements of \( \sum \) with \( n \geq 2 \) such that \( a = {a}_{1}{a}_{2}\cdots {a}_{n} \in \sum \) , then \( \... | Yes |
Lemma 6.11. Let \( \\left( {M,\\Delta }\\right) \) be a pre-Garside monoid and let \( \\sum \\subset M \) be the set of divisors of \( \\Delta \) .\n\n(i) For all \( a, b, c \\in \\sum \), if \( {ac} = {bc} \) or \( {ca} = {cb} \), then \( a = b \) . | Proof. (i) Since \( c \\in \\sum \), there is \( d \\in M \) such that \( {cd} = \\Delta \) . Then \( {ac} = {bc} \) implies \( {a\\Delta } = {acd} = {bcd} = {b\\Delta } \) . Hence \( a = b \) by condition (b) of Definition 6.9. The implication \( {ca} = {cb} \\Rightarrow a = b \) has a similar proof, using an element ... | Yes |
Theorem 6.12. Let \( \\left( {M,\\Delta }\\right) \) be a pre-Garside monoid such that \( M \) is left cancellative.\n\n(i) The monoid \( G \), constructed above, is a group, and the homomorphism \( i : M \\rightarrow G \) is an injection.\n\n(ii) Any element of \( G \) can be written in the form \( i{\\left( \\Delta \... | Proof. (i) If \( i\\left( a\\right) = i\\left( b\\right) \) for \( a, b \\in M \), then \( \\left( {a,0}\\right) \\sim \\left( {b,0}\\right) \) in \( H \). It follows that \( a = {\\Delta }^{0}a = {\\Delta }^{0}b = b \). This proves the injectivity of \( i \).\n\nAny element \( g \\in G \) has the form \( \\pi \\left( ... | Yes |
Theorem 6.14. Let \( \\left( {M,\\Delta }\\right) \) be a pre-Garside monoid such that \( M \) is nontrivial, left cancellative, and atomic. Then any element of \( G = {G}_{M} \\supset M \) can be written uniquely in the form \( {\\Delta }^{s}b \), where \( s \\in \\mathbf{Z} \) and \( b \) is an element of \( M \) tha... | Proof. Note first that \( \\parallel \\Delta \\parallel > 0 \). Indeed, if \( \\parallel \\Delta \\parallel = 0 \), then \( \\Delta = 1 \). Since \( M \) is atomic, the remarks at the end of Section 6.1.3 imply that \( {\\sum }_{\\Delta } = \\{ 1\\} \). Since \( {\\sum }_{\\Delta } \) generates \( M \), we have \( M = ... | Yes |
Lemma 6.16. If \( \left( {M,\Delta }\right) \) is a Garside monoid, then the set \( \sum \) satisfies all conditions of Theorem 6.5. | This key lemma allows us to apply the results of Section 6.2 to Garside monoids. The rest of this subsection is devoted to the proof of Lemma 6.16. We need to verify that \( \sum \) satisfies conditions \( \left( { * }_{1}\right) - \left( { * }_{3}\right) \) of Theorem 6.5. Condition \( \left( { * }_{1}\right) \) direc... | No |
Lemma 6.17. Let \( E \) be a nonempty finite subset of \( M \) satisfying the following two conditions:\n\n(i) if \( a \in M \) and \( b \in E \) with \( a \preccurlyeq b \), then \( a \in E \) ;\n\n(ii) if \( a \in E, s, t \in S \) are such that as, at \( \in E \), then \( a{\Delta }_{s, t} \in E \) .\n\nThen \( E \) ... | Proof. Let \( c \) be an element of \( E \) such that \( \parallel c\parallel \) is maximal (we say that \( c \) is of maximal height in \( E \) ). We wish to show that \( E = \{ a \in M \mid a \preccurlyeq c\} \) . By condition (i), \( \{ a \in M \mid a \preccurlyeq c\} \subset E \) . Let us prove the opposite inclusi... | Yes |
Lemma 6.18. For any \( a, b \in \sum \), the set\n\n\[ E = \{ x \in M \mid x \preccurlyeq a\\text{ and }x \preccurlyeq b\} \\subset \\sum \]\n\nhas a maximal element (with respect to \( \\preccurlyeq \) ). | Proof. Since \( \\sum \) is finite, so is \( E \) . Clearly, \( 1 \\in E \) . The set \( E \) is nonempty since \( 1 \\in E \), and obviously satisfies condition (i) of Lemma 6.17. Let us check condition (ii). We have to show that if \( {xs} \) and \( {xt} \) are left divisors of both \( a \) and \( b \) for some \( s,... | Yes |
Lemma 6.21. An element a of \( {B}_{n}^{ + } \) is reduced if and only if \( \lambda \left( {\pi \left( a\right) }\right) = \ell \left( a\right) \) . | Proof. If \( a = \rho \left( w\right) \) for some \( w \in {\mathfrak{S}}_{n} \), then \( \ell \left( a\right) = \lambda \left( w\right) = \lambda \left( {\pi \left( a\right) }\right) \) . Conversely, let \( a = {\sigma }_{{i}_{1}}\cdots {\sigma }_{{i}_{r}} \in {B}_{n}^{ + } \) with \( r = \ell \left( a\right) = \lambd... | Yes |
Lemma 6.22. A left or right divisor of a reduced element of \( {B}_{n}^{ + } \) is reduced. | Proof. If \( a, b \in {B}_{n} \) and \( {ab} \in {B}_{n}^{\text{red }} \), then\n\n\[ \ell \left( a\right) + \ell \left( b\right) = \ell \left( {ab}\right) = \lambda \left( {\pi \left( {ab}\right) }\right) = \lambda \left( {\pi \left( a\right) \pi \left( b\right) }\right) \leq \lambda \left( {\pi \left( a\right) }\righ... | Yes |
For \( u, v \in {\mathfrak{S}}_{n} \), we have \( \rho \left( u\right) \rho \left( v\right) = \rho \left( {uv}\right) \) if and only if \( \lambda \left( u\right) + \lambda \left( v\right) = \lambda \left( {uv}\right) \) | Proof. Set \( a = \rho \left( u\right) \rho \left( v\right) \in {B}_{n}^{ + } \) . We have \( \pi \left( a\right) = {uv} \) and\n\n\[ \lambda \left( {uv}\right) = \lambda \left( {\pi \left( a\right) }\right) \leq \ell \left( a\right) = \ell \left( {\rho \left( u\right) }\right) + \ell \left( {\rho \left( v\right) }\rig... | Yes |
Lemma 6.24. An element of \( {B}_{n}^{ + } \) is reduced if and only if it is a left (or a right) divisor of \( {\Delta }_{n} \) . | Proof. Since \( {\Delta }_{n} \) is reduced, all its left divisors and right divisors are reduced by Lemma 6.22.\n\nConversely, let \( a = \rho \left( {\pi \left( a\right) }\right) \in {B}_{n}^{\text{red }} \) . Set \( b = \rho \left( {\pi {\left( a\right) }^{-1}{w}_{0}}\right) \in {B}_{n}^{\text{red }}, u = \pi \left(... | Yes |
Theorem 6.25. Any finite family of elements of \( {B}_{n}^{ + } \) has a unique right gcd and a unique left lcm. | Proof. Consider the map rev : \( {B}_{n}^{ + } \rightarrow {B}_{n}^{ + } \) obtained by reading the words in the generators \( {\sigma }_{1},\ldots ,{\sigma }_{n - 1} \) from right to left:\n\n\[ \operatorname{rev}\left( {{\sigma }_{{i}_{1}}{\sigma }_{{i}_{2}}\cdots {\sigma }_{{i}_{r - 1}}{\sigma }_{{i}_{r}}}\right) = ... | Yes |
Theorem 6.30. For any Coxeter matrix \( A \) such that the group \( {W}_{A} \) is finite, the pair \( \left( {{B}_{A}^{ + },\Delta }\right) \) is a comprehensive Garside monoid. | Proof. We have already observed that the monoid \( {B}_{A}^{ + } \) is atomic. Since both sides of (6.8) represent reduced expressions in \( {W}_{A} \), the set \( {B}_{A}^{\text{red }} \) is comprehensive. The proof of conditions (a), (b) of Definition 6.9 reproduces the corresponding part of the proof of Theorem 6.20... | Yes |
For all \( s, t \in S \), the order of \( {\mu }_{s}{\mu }_{t} \) is equal to \( {a}_{s, t} \) . | Proof. (a) If \( {a}_{s, t} = \infty \), then\n\n\[ \left( {{\mu }_{s}{\mu }_{t}}\right) \left( {e}_{s}\right) = {\mu }_{s}\left( {{e}_{s} + 2{e}_{t}}\right) = - {e}_{s} + 2\left( {{e}_{t} + 2{e}_{s}}\right) = 3{e}_{s} + 2{e}_{t} \]\n\nand\n\n\[ \left( {{\mu }_{s}{\mu }_{t}}\right) \left( {e}_{t}\right) = - {\mu }_{s}\... | Yes |
Theorem 6.32 (E. Brieskorn). The map \( S \rightarrow {\pi }_{1}\left( {W \smallsetminus E,\dot{p}}\right), s \mapsto \dot{s} \) induces a group isomorphism \( {B}_{A} \cong {\pi }_{1}\left( {W \smallsetminus E,\dot{p}}\right) \) . | For a proof, see Brieskorn [Bri71] or Deligne [Del72]. These authors also proved that the manifold \( W \smallsetminus E \) is aspherical, i.e., its higher homotopy groups vanish, see [Del72],[Bri73]. Since \( E \rightarrow W \smallsetminus E \) is a covering, the manifold \( E \) is also aspherical. Its fundamental gr... | No |
Lemma 7.1. (a) If \( {G}_{1},\ldots ,{G}_{r} \) are orderable groups, then their direct product \( {G}_{1} \times \cdots \times {G}_{r} \) is orderable. | Proof. (a) Let \( { \leq }_{i} \) be a left-invariant total order on \( {G}_{i} \) . We define a relation \( \leq \) on \( G = {G}_{1} \times \cdots \times {G}_{r} \) by \( \left( {{x}_{1},\ldots ,{x}_{r}}\right) \leq \left( {{y}_{1},\ldots ,{y}_{r}}\right) \) if either \( {x}_{i} = {y}_{i} \) for all \( i \in \{ 1,\ld... | Yes |
For any subset \( \mathcal{P} \) of a group \( G \), \[ \mathcal{P} \cap \{ 1\} = \varnothing \Leftrightarrow {\mathcal{P}}^{-1} \cap \{ 1\} = \varnothing \Leftarrow \mathcal{P} \cap {\mathcal{P}}^{-1} = \varnothing . \] If \( {\mathcal{P}}^{2} \subset \mathcal{P} \), then \( \mathcal{P} \cap \{ 1\} = \varnothing \Righ... | Proof. If \( 1 \in \mathcal{P} \), then \( 1 = {1}^{-1} \in {\mathcal{P}}^{-1} \) . This shows that \( {\mathcal{P}}^{-1} \cap \{ 1\} = \varnothing \Rightarrow \) \( \mathcal{P} \cap \{ 1\} = \varnothing \) . Replacing here \( \mathcal{P} \) by \( {\mathcal{P}}^{-1} \), we obtain the converse implication. To prove the ... | Yes |
Lemma 7.3. Let \( \leq \) be a left-invariant order on a group \( G \). Set\n\n\[ \mathcal{P} = \{ x \in G \mid x > 1\} .\n\]\n\nThen \( {\mathcal{P}}^{-1} = \{ x \in G \mid x < 1\} ,{\mathcal{P}}^{2} \subset \mathcal{P} \), and\n\n\[ \mathcal{P} \cap \{ 1\} = {\mathcal{P}}^{-1} \cap \{ 1\} = \mathcal{P} \cap {\mathcal... | Proof. If \( x \in {\mathcal{P}}^{-1} \), then \( {x}^{-1} \in \mathcal{P} \), so that \( 1 < {x}^{-1} \). Multiplying by \( x \) on the left, we obtain \( x < 1 \). Similarly, \( x < 1 \) implies \( 1 = {x}^{-1}x < {x}^{-1}1 = {x}^{-1} \), so that \( {x}^{-1} \in \mathcal{P} \) and \( x \in {\mathcal{P}}^{-1} \). This... | Yes |
Theorem 7.4. Let \( \mathcal{P} \) be a subset of a group \( G \) such that\n\n\[{\mathcal{P}}^{2} \subset \mathcal{P}\;\text{ and }\;1 \notin \mathcal{P}.\]\n\nThen \( G \) has a unique left-invariant order \( \leq \) such that \( \mathcal{P} = \{ x \in G \mid x > 1\} \) . If \( z\mathcal{P}{z}^{-1} \subset \mathcal{P... | Proof. Let us first prove the uniqueness of the order. By the left invariance, the inequality \( x < y \) is equivalent to the inequality \( 1 = {x}^{-1}x < {x}^{-1}y \) . The latter is equivalent to the inclusion \( {x}^{-1}y \in \mathcal{P} \) . This shows that a left-invariant order on \( G \) with positive cone \( ... | Yes |
Proposition 7.5. Any orderable group \( G \) is torsion free. | Proof. We have to show that \( {x}^{n} \neq 1 \) for any integer \( n \geq 1 \) and any \( x \in G \) such that \( x \neq 1 \) . Suppose that \( x > 1 \) . Then by the left invariance,\n\n\[ \n{x}^{n} = \left( {x}^{n - 1}\right) x > {x}^{n - 1}1 = {x}^{n - 1} \n\]\n\nfor any \( n \geq 1 \) . By induction, \( {x}^{n} > ... | Yes |
Proposition 7.6. If \( G \) is an orderable group and \( R \) is a ring without zero-divisors, then the group algebra \( R\left\lbrack G\right\rbrack \) has no zero-divisors. | Proof. Let \( \omega = \mathop{\sum }\limits_{{i = 1}}^{p}{r}_{i}{g}_{i} \) and \( {\omega }^{\prime } = \mathop{\sum }\limits_{{j = 1}}^{q}{s}_{j}{h}_{j} \) be nonzero elements of \( R\left\lbrack G\right\rbrack \) , where \( {g}_{1},\ldots ,{g}_{p} \) and \( {h}_{1},\ldots ,{h}_{q} \) are elements of \( G \), and \( ... | Yes |
Lemma 7.7. Let \( G \) be a biorderable group. Then \( {x}^{n} = {y}^{n} \Rightarrow x = y \) for any \( x, y \in G \) and any positive integer \( n \) . | Proof. We start with the following observation: in a biorderable group, \( x < y \) together with \( {x}^{\prime } < {y}^{\prime } \) implies \( x{x}^{\prime } < y{y}^{\prime } \) . Indeed, by the left and right invariance of the order, \( x{x}^{\prime } < x{y}^{\prime } < {x}^{\prime }{y}^{\prime } \) . From this an e... | Yes |
Theorem 7.8. The pure braid group \( {P}_{n} \) is biorderable for all \( n \geq 1 \) . | To prove this theorem we first study Magnus expansions of free groups and then show that free groups are biorderable. Theorem 7.8 is proven in Section 7.2.3. Neither this theorem nor its proof will be used in the sequel. | No |
Proposition 7.10. Let \( F \) be a free group freely generated by a set \( X \) . The homomorphism of groups \( \mu : F \rightarrow G\left( X\right) \) defined by \( \mu \left( x\right) = 1 + x \) for all \( x \in X \) is injective. | Proof. The existence and the uniqueness of \( \mu \) follow from the definition of \( F \) . To check the injectivity of \( \mu \), pick a nontrivial element \( w \in F \) and write it in the form\n\n\[ w = {x}_{1}^{{k}_{1}}{x}_{2}^{{k}_{2}}\cdots {x}_{r}^{{k}_{r}} \]\n\nwhere \( {x}_{1},{x}_{2},\ldots ,{x}_{r} \in X \... | Yes |
Lemma 7.14. The subset of \( {B}_{n} \) consisting of all \( \sigma \) -negative elements is \( {\mathcal{P}}^{-1} \), and \( {\mathcal{P}}^{2} \subset \mathcal{P} \). | Proof. (a) Let \( \beta \) be a \( \sigma \) -negative element of \( {B}_{n} \). Then it can be represented by a \( \sigma \) -negative braid word \( w \). By definition, the inverse word \( {w}^{-1} \) is \( \sigma \) -positive. It represents \( {\beta }^{-1} \in {B}_{n} \). Then \( {\beta }^{-1} \in \mathcal{P} \) an... | Yes |
Proposition 7.18. (a) \( {\sigma }_{n - 1} \) is the smallest \( \sigma \) -positive element of \( {B}_{n} \) . | Proof. (a) Suppose that there is \( \beta \in \mathcal{P} \) such that \( \beta < {\sigma }_{n - 1} \) ; this is equivalent to \( {\beta }^{-1}{\sigma }_{n - 1} \in \mathcal{P} \) . Let \( w \) be a \( {\sigma }_{i} \) -positive word representing \( \beta \) . The braid \( {\beta }^{-1}{\sigma }_{n - 1} \) is represent... | Yes |
Proposition 7.19. The intervals \( {\left\{ \beta \in {B}_{n} \mid {\Delta }_{n}^{2k} \leq \beta < {\Delta }_{n}^{2\left( {k + 1}\right) }\right\} }_{k \in \mathbf{Z}} \) form a partition of \( {B}_{n} \) . | Proof. Since \( {\Delta }_{n}^{2} \) belongs to \( {B}_{n}^{ + } \), we have \( {\Delta }_{n}^{2} > 1 \) . Hence,\n\n\[ \cdots < {\Delta }_{n}^{-6} < {\Delta }_{n}^{-4} < {\Delta }_{n}^{-2} < 1 < {\Delta }_{n}^{2} < {\Delta }_{n}^{4} < {\Delta }_{n}^{6} < \cdots . \]\n\nTo prove the proposition it therefore suffices to... | Yes |
Proposition 7.22. There is a unique left-invariant total order on \( {B}_{\infty } \) such that the inclusions \( {B}_{n} \hookrightarrow {B}_{\infty } \) are order-preserving. As an ordered set, \( {B}_{\infty } \) is isomorphic to the ordered set \( \mathbf{Q} \) of rational numbers. | Proof. (a) Let \( \beta ,{\beta }^{\prime } \in {B}_{\infty } \) . By definition, there is \( n \) such that \( \beta ,{\beta }^{\prime } \in {B}_{n} \) . We set \( \beta { \leq }_{\infty }{\beta }^{\prime } \) if \( \beta { \leq }_{n}{\beta }^{\prime } \) . It follows from Lemma 7.21 that this is independent of the ch... | No |
Lemma 7.23. (a) \( T\left( {\widetilde{\sigma }}_{i}\right) = {\widetilde{\sigma }}_{i + 1} \) for all \( i \in \{ 1,\ldots, n - 2\} \) . | Proof. (a) This follows from the definitions. | No |
Lemma 7.26. A braid word of index \( i \in \{ 1,\ldots, n - 1\} \) that does not contain \( {\sigma }_{i} \) -handles is \( {\sigma }_{i} \) -positive or \( {\sigma }_{i} \) -negative. | A concrete way to visualize the \( {\sigma }_{i} \) -handles contained in a braid word \( w \) is to delete from \( w \) all occurrences of \( {\sigma }_{j}^{\pm 1} \) with \( j > i \), thus obtaining a possibly shorter word \( w\left\lbrack i\right\rbrack \) . The braid word \( w \) contains a \( {\sigma }_{i} \) -han... | No |
Lemma 7.28. (a) A prime handle contains no other handles.\n\n(b) Any braid word containing at least one handle contains a unique prime handle. | Proof. (a) Let \( w = {w}_{1}v{w}_{2} \) be a braid word in which \( v \) is a prime handle. Suppose that \( v = {w}^{\prime }u{w}^{\prime \prime } \), where \( u \) is a handle. Then \( {w}_{1}{w}^{\prime }u \) is a prefix of \( w \) containing a handle. Since \( {w}_{1}v = {w}_{1}{w}^{\prime }u{w}^{\prime \prime } \)... | Yes |
Lemma 7.31. Any handle represents the same element of \( {B}_{n} \) as its reduction. | Proof. This is a consequence of the relations\n\n\[ \n{\sigma }_{i}^{e}{\sigma }_{j}^{\pm 1}{\sigma }_{i}^{-e} = \left\{ \begin{array}{ll} {\sigma }_{j}^{\pm 1} & \text{ if }j \geq i + 2, \\ {\sigma }_{i + 1}^{-e}{\sigma }_{i}^{\pm 1}{\sigma }_{i + 1}^{e} & \text{ if }j = i + 1, \end{array}\right. \n\]\n\nwhich follow ... | Yes |
Lemma 7.35. Let \( {N}_{r} \) be the number of edges of \( {\Gamma }_{r} \) . For \( i \in \{ 1,\ldots, n - 1\} \) , any \( {\sigma }_{i} \) -positive (resp. \( {\sigma }_{i} \) -negative) braid word that is the label of a path in \( {\Gamma }_{r} \) contains the letter \( {\sigma }_{i} \) (resp. \( {\sigma }_{i}^{-1} ... | Proof. We give the proof for the \( {\sigma }_{i} \) -positive case. The \( {\sigma }_{i} \) -negative case can be treated in a similar way.\n\nLet \( a = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{k}}\right) \) be a path in \( {\Gamma }_{r} \) whose label is a \( {\sigma }_{i} \) -positive word \( w \) . The word \( w \) ha... | Yes |
Lemma 7.36. For any braid word \( w \), there is an integer \( r \geq 0 \) and a path in \( {\Gamma }_{r} \) whose label is \( w \) . | Proof. Let \( a = \left( {{a}_{1},{a}_{2},\ldots ,{a}_{k}}\right) \) be a path in \( \Gamma \) with label \( w \) . We denote the initial vertex of \( {a}_{1} \) by \( {\beta }_{0} \) and the terminal vertex of \( {a}_{i} \) by \( {\beta }_{i}\left( {1 \leq i \leq k}\right) \) . By definition of a path, the initial ver... | No |
Lemma 7.37. If \( a \) is a path in \( {\Gamma }_{r} \) with \( r \geq 0 \), then so is \( \operatorname{red}\left( a\right) \) . | The proof of this lemma given in Section 7.5.6 is based on the following decomposition of prime handle reductions into elementary steps.\n\nLet \( w,{w}^{\prime } \) be braid words. We say that \( {w}^{\prime } \) is obtained from \( w \) by an elementary reduction if \( {w}^{\prime } \) is obtained from \( w \) by rep... | Yes |
Lemma 7.39. Let \( w \) be a braid word of index \( i \) containing at least one handle. Assume that \( w \) is the label of a path a in \( {\Gamma }_{r} \) with initial vertex \( {\beta }_{0} \) . Then \( h\left( {\operatorname{red}\left( w\right) }\right) \leq h\left( w\right) \) . If \( h\left( {\operatorname{red}\l... | Proof. If \( h\left( w\right) = 0 \), then \( w \) contains no \( {\sigma }_{i} \) -handles, and one passes from \( w \) to \( \operatorname{red}\left( w\right) \) by reducing some \( {\sigma }_{j} \) -handle with \( j > i \) . It is clear that \( \operatorname{red}\left( w\right) \) contains no \( {\sigma }_{i} \) -ha... | Yes |
Proposition 7.41. Let \( G \) be a group acting on a totally ordered set \( X \) by order-preserving bijections such that there is an element of \( X \) whose stabilizer is trivial. Then \( G \) is orderable. | Proof. For \( f \in G \) and \( b \in X \), let \( f\left( b\right) \in X \) be the result of the action of \( f \) on \( b \) . By assumption, \( b < {b}^{\prime } \Rightarrow f\left( b\right) < f\left( {b}^{\prime }\right) \) in \( X \) for all \( b,{b}^{\prime } \in X \) and \( f \in G \) , and there is \( a \in X \... | Yes |
Example 1.2. Let \( s : {\mathbb{R}}^{n} \rightarrow {\mathbb{R}}^{n} \) interchange the first two coordinates, i.e., \[ s\left( {{x}_{1},{x}_{2},{x}_{3},\ldots ,{x}_{n}}\right) = \left( {{x}_{2},{x}_{1},{x}_{3},\ldots ,{x}_{n}}\right) . \] Equivalently, \( s \) transposes the first two standard basis vectors \( {e}_{1... | For future reference, we derive a formula for \( {s}_{\alpha } \) . Given \( x \in V \), write \( x = \) \( h + {\lambda \alpha } \) with \( h \in H \) and \( \lambda \in \mathbb{R} \) . Taking the inner product of both sides with \( \alpha \) , we obtain \( \lambda = \langle \alpha, x\rangle /\langle \alpha ,\alpha \r... | Yes |
Lemma 1.4. Let \( \Phi \) be a finite set of nonzero vectors in \( V \), and let \( W \) be the group generated by the reflections \( {s}_{\alpha }\left( {\alpha \in \Phi }\right) \) . If \( \Phi \) is invariant under the action of \( W \), then \( W \) is finite. | Proof. We will show that \( W \) is isomorphic to a group of permutations of the finite set \( \Phi \) . Let \( {V}_{1} \) be the subspace of \( V \) spanned by \( \Phi \), and let \( {V}_{0} \) be its orthogonal complement. Then \( {V}_{0} = \mathop{\bigcap }\limits_{{\alpha \in \Phi }}{\alpha }^{ \bot } \), which is ... | Yes |
Let \( V \) be 2-dimensional, and choose two hyperplanes (lines) that intersect at an angle of \( \pi /m \) for some integer \( m \geq 2 \) . Let \( s \) and \( t \) be the corresponding reflections and let \( W \) be the group \( \langle s, t\rangle \) they generate. Then the product \( \rho \mathrel{\text{:=}} {st} \... | This group \( W \) is called the dihedral group of order \( {2m} \), and we will denote it by \( {D}_{2m} \) . If \( m \geq 3, W \) is the group of symmetries of a regular \( m \) -gon. If \( m = 3,4 \), or 6, then \( W \) can also be described as the Weyl group of a root system \( \Phi \), said to be of type \( {\math... | Yes |
Let \( W \) be the group of linear transformations of \( {\mathbb{R}}^{n}\left( {n \geq 2}\right) \) that permute the standard basis vectors \( {e}_{1},{e}_{2},\ldots ,{e}_{n} \) . Thus \( W \) is isomorphic to the symmetric group \( {S}_{n} \) on \( n \) letters and can be identified with the group of \( n \times n \)... | The interested reader can verify that \( W \) is the group of symmetries of a regular \( \left( {n - 1}\right) \) -simplex in \( {V}_{1} \) . [Hint: The convex hull \( \sigma \) of \( {e}_{1},\ldots ,{e}_{n} \) is a regular \( \left( {n - 1}\right) \) -simplex in the affine hyperplane \( \sum {x}_{i} = 1 \), which is p... | No |
Proposition 1.24. Let \( A \) be an open cell.\n\n(1) \( \bar{A} \) is the closure of \( A \) in \( V \) (in the sense of point-set topology).\n\n(2) Let \( L \) be the linear span of \( \bar{A} \) . Then \( A \) is the interior of \( \bar{A} \) in \( L \), i.e., the largest open subset of \( L \) contained in \( \bar{... | Proof. (1) Clearly \( \bar{A} \) is closed in \( V \), so it contains the closure of \( A \) . Conversely, given \( y \in \bar{A} \), choose \( x \in A \) and consider the closed line segment from \( x \) to \( y \) , denoted by \( \left\lbrack {x, y}\right\rbrack \) . Each equality in the description of \( A \) holds ... | Yes |
Proposition 1.25. Let \( X \) be a set defined by equalities or weak inequalities as above. Then \( X \) is a closed cell with respect to \( \mathcal{H} \) . | Proof. Let \( {\sigma }_{i} \) be 0 if \( {f}_{i} = 0 \) on \( X \) . Otherwise, either \( {f}_{i} \geq 0 \) on \( X \) or \( {f}_{i} \leq 0 \) on \( X \), and we take \( {\sigma }_{i} \) to be + or -, accordingly. [Caution: It is possible that our original description of \( X \) involved an inequality, say \( {f}_{i} ... | Yes |
Proposition 1.27. Let \( A \) be a cell. Then two distinct points \( y, z \in \bar{A} \) lie in the same face of \( A \) if and only if there is an open line segment containing both \( y \) and \( z \) and lying entirely in \( \bar{A} \) . Consequently, the partition of \( \bar{A} \) into faces depends only on \( A \) ... | Proof. Suppose \( y \) and \( z \) are in the same face \( B \leq A \) . For each condition \( {f}_{i} = {\sigma }_{i} \) in the description of \( B \), we can extend the segment \( \left\lbrack {y, z}\right\rbrack \) slightly in both directions without violating the condition. Since there are only finitely many such c... | Yes |
Lemma 1.31. If \( H \in \mathcal{H} \) is not a wall of \( C \), then \( C \) is defined by \( {\mathcal{H}}^{\prime } \mathrel{\text{:=}} \) \( \mathcal{H} \smallsetminus \{ H\} \) . | Proof. Assume, to simplify the notation, that \( C \) is defined by the inequalities \( {f}_{i} > 0 \) for all \( i \), and let \( j \) be the index such that \( H = {H}_{j} \) . Suppose \( C \) is not defined by \( {\mathcal{H}}^{\prime } \) . Then removing the inequality \( {f}_{j} > 0 \) results in a set \( {C}^{\pr... | Yes |
Proposition 1.32. Let \( C \) be a chamber and let \( {\mathcal{H}}_{C} \) be its set of walls. Then \( C \) is defined by \( {\mathcal{H}}_{C} \), and \( {\mathcal{H}}_{C} \) is the smallest subset of \( \mathcal{H} \) with this property. | Proof. If \( C \) is defined by \( {\mathcal{H}}^{\prime } \subseteq \mathcal{H} \), then we can use \( {\mathcal{H}}^{\prime } \) to determine the walls of \( C \) by Corollary 1.28; hence \( {\mathcal{H}}^{\prime } \supseteq {\mathcal{H}}_{C} \) . It remains to show that \( C \) is defined by \( {\mathcal{H}}_{C} \) ... | Yes |
Proposition 1.33. Let \( C \) be a chamber and let \( H \) be a linear hyperplane in \( V \) . Then \( H \) is a wall of \( C \) if and only if \( C \) lies on one side of \( H \) and \( \bar{C} \cap H \) has nonempty interior in \( H \) . | Proof. If \( H \) is the support of a panel \( A \) of \( C \), then certainly \( C \) lies on one side of \( H \) and \( \bar{C} \cap H \) contains \( A \), which is a nonempty open subset of \( H \) . Conversely, suppose \( H \) is a hyperplane such that \( C \) lies on one side of \( H \) and \( \bar{C} \cap H \) ha... | Yes |
Proposition 1.36. Assume that \( \mathcal{H} \) is essential. Then the following conditions on the chamber \( C \) are equivalent:\n\n(i) \( C \) is a simplicial cone.\n\n(ii) \( C \) has exactly \( n \) panels, i.e., \( r = n \) .\n\n(iii) \( {f}_{1},\ldots ,{f}_{r} \) are linearly independent.\n\n(iv) \( {f}_{1},\ldo... | Proof. As we noted above, the assumption that \( \mathcal{H} \) is essential implies that \( \mathop{\bigcap }\limits_{{i = 1}}^{r}{H}_{i} = 0 \), i.e., that the equations \( {f}_{1} = 0,\ldots ,{f}_{r} = 0 \) have only the trivial solution. The equivalence of (ii), (iii), and (iv) follows easily from this by elementar... | No |
Proposition 1.37. Assume that \( \mathcal{H} \) is essential. If \( \left\langle {{e}_{i},{e}_{j}}\right\rangle \leq 0 \) for each \( i \neq j \) \( \left( {i, j \leq r}\right) \), i.e., if the angle between \( {e}_{i} \) and \( {e}_{j} \) is not acute, then \( C \) is a simplicial cone. | Proof. According to Proposition 1.36, we must show that \( {e}_{1},\ldots ,{e}_{r} \) are linearly independent. If not, we claim that there is a nontrivial linear relation among them with nonnegative coefficients. For let \( \mathop{\sum }\limits_{{i = 1}}^{r}{\lambda }_{i}{e}_{i} = 0 \) be an arbitrary nontrivial line... | Yes |
Proposition 1.40. Given a cell \( A \) and a chamber \( C \), the product \( {AC} \) (or the projection of \( C \) on \( A \) ) is a chamber having \( A \) as a face; among the chambers having \( A \) as a face, it is the unique one at minimal distance from \( C \) . | Proof. To minimize the distance to \( C \) of a chamber \( D \geq A \), we must maximize the number of indices \( i \) such that \( {\sigma }_{i}\left( D\right) = {\sigma }_{i}\left( C\right) \) . We have no choice about \( {\sigma }_{i}\left( D\right) \) whenever \( {\sigma }_{i}\left( A\right) \neq 0 \), so the best ... | Yes |
Proposition 1.47. Any two elements of \( \sum \) have a greatest lower bound. | We will denote by \( A \cap B \) the greatest lower bound of two open cells \( A \) and \( B \) . It is, of course, not the set-theoretic intersection of \( A \) and \( B \), this intersection being empty unless \( A = B \) ; it is, rather, the open cell whose closure is the intersection of \( \bar{A} \) and \( \bar{B}... | Yes |
Proposition 1.48. Any cell \( A \in \sum \) is a face of a chamber. If \( A \) is a panel, then it is a face of exactly two chambers. | Proof. Choose an arbitrary chamber \( C \) . [Such a \( C \) certainly exists: \( V \) is not the union of finitely many hyperplanes.] Then the projection \( {AC} \) defined in Section 1.4.6 is a chamber having \( A \) as a face. If \( A \) is a panel with support \( {H}_{i} \) , then the sign sequence of a chamber \( ... | Yes |
Corollary 1.49. Every \( H \in \mathcal{H} \) is a wall of a chamber. | Proof. \( H \) cannot be the union of its intersections with the other hyperplanes, so there is at least one panel \( A \) with support \( H \) . Hence \( H \) is a wall of each of the chambers \( C > A \) . | Yes |
Lemma 1.55. For any two chambers \( C \neq D \), there is a chamber \( {C}^{\prime } \) adjacent to \( C \) such that \( {d}_{\mathcal{H}}\left( {{C}^{\prime }, D}\right) = {d}_{\mathcal{H}}\left( {C, D}\right) - 1 \) . | Proof. Since \( C \) is defined by its set of walls (Proposition 1.32), there must be a wall of \( C \) that separates \( C \) from \( D \) . [Otherwise, we would have \( D \subseteq C \) , contradicting the fact that distinct chambers are disjoint.] Let \( A \) be the corresponding panel of \( C \), and let \( {C}^{\p... | Yes |
Proposition 1.57. The diameter of \( \mathcal{C} \) is \( m \mathrel{\text{:=}} \left| \mathcal{H}\right| \) . For any chamber \( C \) , there is a unique chamber \( D \) with \( d\left( {C, D}\right) = m \), namely, the opposite chamber \( D = - C \) . | Observe that for any chambers \( C \) and \( D \) , \[ d\left( {C, D}\right) + d\left( {D, - C}\right) = m. \] Indeed, every hyperplane in \( \mathcal{H} \) separates \( D \) from either \( C \) or \( - C \), but not both. Thus if we concatenate a minimal gallery from \( C \) to \( D \) with a minimal gallery from \( D... | Yes |
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