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Corollary 5.6.15. \( \left( {G = \mathbf{{Sp}}\left( {n,\mathbb{C}}\right) }\right) \) In the canonical duality decomposition\n\n\[ \mathcal{P}\left( V\right) \cong {\bigoplus }_{\lambda \in \mathcal{S}}{E}^{\lambda } \otimes {F}^{\lambda } \]\n\n(5.96)\n\nunder the joint action of \( \mathbb{D}{\left( V\right) }^{G} \... | Proof. Apply Corollary 5.6.4. | No |
Theorem 5.7.1. Suppose \( T \in \mathbb{D}{\left( V\right) }^{G} \) . Then there exists \( z \in Z\left( \mathfrak{g}\right) \) such that \( T = \) \( {d\tau }\left( z\right) \) . Hence \( \mathbb{D}{\left( V\right) }^{G} \) is commutative and \( \mathcal{P}\left( V\right) \) is a multiplicity-free G-module. | The theorem will follow from Theorem 4.2.13 once we show that \( \mathbb{D}{\left( V\right) }^{G} \subset \) \( {d\tau }\left( {U\left( \mathfrak{g}\right) }\right) \) . For this, we will use the tensor form of the FFT for \( G \) . If \( x \in V \) and \( \xi \in {V}^{ * } \) then\n\n\[ x = \mathop{\sum }\limits_{{i, ... | No |
Theorem 5.7.3. The space of homogeneous polynomials on \( S{M}_{n} \) of degree \( r \) decomposes under \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) as\n\n\[ \n{\mathcal{P}}^{r}\left( {S{M}_{n}}\right) \cong {\bigoplus }_{\mu }{F}_{n}^{\mu }\n\]\n\nwith the sum over all nonnegative dominant weights \( \mu = \mathop... | Proof. We follow the same line of argument as in Theorem 5.6.7. In this case the algebra \( \mathcal{P}\left( {S{M}_{n}}\right) \) is generated by the matrix entry functions \( {x}_{ij} \) with \( i \leq j \), and\n\n\[ \n\rho \left( h\right) {x}_{ij} = {h}_{i}{h}_{j}{x}_{ij}\;\text{ for }h = \operatorname{diag}\left\l... | Yes |
Lemma 6.1.4. Suppose \( \left( {{S}^{\prime },{\gamma }^{\prime }}\right) \) is a space of spinors for \( \left( {V,\beta }\right) \) . Let \( n = \dim V \) .\n\n1. Set \( Z = \mathop{\bigcap }\limits_{{w \in W}}\operatorname{Ker}{\gamma }^{\prime }\left( w\right) \) . Then \( Z \neq 0 \) . | Proof. Take a basis \( {e}_{\pm i} \) for \( V \) as above. We have\n\n\[ Z = \mathop{\bigcap }\limits_{{i = 1}}^{l}\operatorname{Ker}{\gamma }^{\prime }\left( {e}_{i}\right) \]\n\n(where \( n = {2l} \) or \( {2l} + 1 \) ). Now \( {\gamma }^{\prime }{\left( {e}_{1}\right) }^{2} = 0 \), since \( \beta \left( {{e}_{1},{e... | Yes |
Proposition 6.1.5. Suppose \( \dim V = n \) is even. Let \( \left( {S,\gamma }\right) \) be a space of spinors for \( \left( {V,\beta }\right) \) . Then \( \left( {\operatorname{End}S,\gamma }\right) \) is a Clifford algebra for \( \left( {V,\beta }\right) \) . Thus \( \operatorname{Cliff}\left( {V,\beta }\right) \) is... | Proof. Let \( \widetilde{\gamma } : \operatorname{Cliff}\left( {V,\beta }\right) \rightarrow \) End \( S \) be the canonical algebra homomorphism extending the map \( \gamma : V \rightarrow \) End \( S \) . Since \( \widetilde{\gamma } \) is an irreducible representation, we know by Corollary 4.1.7 that \( \widetilde{\... | Yes |
Proposition 6.1.6. Suppose \( \dim V = {2l} + 1 \) is odd. Let \( \left( {S,{\gamma }_{ + }}\right) \) and \( \left( {S,{\gamma }_{ - }}\right) \) be the two inequivalent spaces of spinors for \( \left( {V,\beta }\right) \), and let \( \gamma : V \rightarrow \operatorname{End}S \oplus \operatorname{End}S \) be defined ... | Proof. Let \( l \geq 1 \) (the case \( \dim V = 1 \) is left to the reader) and use the model \( S = \) \( {U}_{1} \otimes \cdots \otimes {U}_{l} \) for spinors, with \( {\gamma }_{ \pm }\left( {e}_{\pm i}\right) = {A}_{\pm i} \) and \( {\gamma }_{ \pm }\left( {e}_{0}\right) = \pm {A}_{0} \) (notation as above). Let\n\... | Yes |
Lemma 6.1.7. There is an algebra isomorphism \( \operatorname{Cliff}\left( {{V}_{0},{\beta }_{0}}\right) \cong {\operatorname{Cliff}}^{ + }\left( {V,\beta }\right) \) . Hence \( {\operatorname{Cliff}}^{ + }\left( {V,\beta }\right) \) is a simple algebra. | Proof. Let \( \gamma : V \rightarrow \operatorname{Cliff}\left( {V,\beta }\right) \) be the canonical map. For \( v \in {V}_{0} \) we define \( \varphi \left( v\right) = \) i \( \gamma \left( {e}_{0}\right) \gamma \left( v\right) \) . Then \( \varphi \left( v\right) \in {\operatorname{Cliff}}^{ + }\left( {V,\beta }\rig... | Yes |
Lemma 6.2.1. The linear transformations \( {R}_{a, b} \), with \( a, b \) ranging over \( V \), span \( \mathfrak{{so}}\left( {V,\beta }\right) \) . | Proof. First consider the case in which \( V = W \oplus {W}^{ * } \) has dimension \( {2l} \), where \( W \) and \( {W}^{ * } \) are maximal isotropic. For \( x, y \in W \) and \( {x}^{ * },{y}^{ * } \in {W}^{ * } \), we have\n\n\[ \n{R}_{x,{x}^{ * }}\left( {y + {y}^{ * }}\right) = \left\langle {{x}^{ * }, y}\right\ran... | Yes |
Proposition 6.2.3. ( \( \dim V = {2l} \) ) The half-spin representations \( {\pi }^{ \pm } \) of \( \mathfrak{{so}}\left( {V,\beta }\right) \) are irreducible with highest weights \( {\varpi }_{ \pm } = \left( {{\varepsilon }_{1} + \cdots + {\varepsilon }_{l - 1} \pm {\varepsilon }_{l}}\right) /2 \) . The weights are \... | Proof. Take a \( \beta \) -isotropic basis \( {e}_{\pm i} \) as in Section 2.4.1, and for each ordered index \( I = \left\{ {1 \leq {i}_{1} < \cdots < {i}_{p} \leq l}\right\} \) set \( {u}_{I} = {e}_{-{i}_{1}} \land \cdots \land {e}_{-{i}_{p}} \) (with \( {u}_{\varnothing } = 1 \) ). The diagonal subalgebra \( \mathfra... | Yes |
Proposition 6.2.4. \( \left( {\dim V = {2l} + 1}\right) \) The spin representation of \( \mathfrak{{so}}\left( {V,\beta }\right) \) is irreducible and has highest weight \( {\varpi }_{l} = \left( {{\varepsilon }_{1} + \cdots + {\varepsilon }_{l - 1} + {\varepsilon }_{l}}\right) /2 \) . The weights are\n\n\[ \left( {\pm... | Proof. From the definition of \( {\gamma }_{ + } \), we see that the diagonal subalgebra of \( \mathfrak{{so}}\left( {V,\beta }\right) \) has the same action as in the even-dimensional case treated in Proposition 6.2.3. Thus the weights are given by (6.17) and have multiplicity one. The only dominant weight is \( {\var... | Yes |
Lemma 6.3.1. Suppose \( x, y \in V \) and \( \beta \left( {x, x}\right) = \beta \left( {y, y}\right) \neq 0 \). 1. If \( x - y \) is nonisotropic, then \( {s}_{x - y}x = y \). 2. If \( x - y \) is isotropic, then \( x + y \) is nonisotropic and \( {s}_{y}{s}_{x + y}x = y \). 3. \( \mathbf{O}\left( {V,\beta }\right) \) ... | Proof. We may assume that \( \beta \left( {x, x}\right) = 1 \) . Obviously, \[ {2\beta }\left( {x, x - y}\right) = \beta \left( {x - y, x - y}\right) , \] since \( \beta \left( {x, x}\right) = \beta \left( {y, y}\right) \) . This implies (1). Now assume that \( x - y \) is isotropic. Then \[ {2\beta }\left( {x, y}\righ... | Yes |
Theorem 6.3.5. The group \( \operatorname{Spin}\left( {V,\beta }\right) \) is the identity component of \( \operatorname{Pin}\left( {V,\beta }\right) \), and the homomorphism \( \pi : \operatorname{\mathbf{S} \mathbf{p} \mathbf{i} \mathbf{n} }\left( {V,\beta }\right) \rightarrow \operatorname{\mathbf{S} \mathbf{O} }\le... | Proof. Since a reflection has determinant \( - 1,\mathbf{{SO}}\left( {V,\beta }\right) \) is generated by products of an even number of reflections. By Corollary 6.3.3, \( \operatorname{\mathbf{S} \mathbf{p} \mathbf{i} \mathbf{n} }\left( {V,\beta }\right) \) is generated by \( \pm 1 \) and products of an even number of... | Yes |
Theorem 6.3.6. The Lie algebra of \( \operatorname{\mathbf{S} \mathbf{p} \mathbf{i} \mathbf{n} }\left( {V,\beta }\right) \) is \( \varphi \left( {\operatorname{\mathfrak{s}\mathfrak{o}}\left( {V,\beta }\right) }\right) \), where \( \varphi \) is the isomorphism of Lemma 6.2.2. | Proof. Since \( \operatorname{Spin}\left( {V,\beta }\right) \) is a subgroup of the invertible elements of \( \operatorname{Cliff}\left( {V,\beta }\right) \), we may identify \( \operatorname{Lie}\left( {\operatorname{Spin}\left( {V,\beta }\right) }\right) \) with a Lie subalgebra \( \mathfrak{g} \) of \( \operatorname... | Yes |
Corollary 6.3.7. Let \( P \) be the weight lattice of \( \mathfrak{{so}}\left( {V,\beta }\right) \). For \( \lambda \in {P}_{+ + } \) there is an irreducible regular representation of \( \operatorname{\mathbf{S} \mathbf{p} \mathbf{i} \mathbf{n} }\left( {V,\beta }\right) \) with highest weight \( \lambda \). | Proof. The spin representation (when \( \dim V \) is odd) or half-spin representations (when \( \dim V \) is even) furnish the fundamental representations not obtainable from \( \bigwedge V \) (cf. Theorem 5.5.13). Every regular representation of \( \mathbf{{SO}}\left( {V,\beta }\right) \) gives a regular representatio... | Yes |
Theorem 6.3.8. Let \( G \) be a connected linear algebraic group and let \( \mathfrak{g} = \operatorname{Lie}\left( G\right) \). Suppose every finite-dimensional representation of \( \mathfrak{g} \) is the differential of a regular representation of \( G \). Then \( G \) is algebraically simply connected. | Proof. Let \( \pi : H \rightarrow G \) be a covering homomorphism, where \( H \subset \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) is a connected linear algebraic subgroup. We shall use a general result about linear algebraic groups that will be proved in Chapter 11; namely, that \( H \) is a connected Lie group (Theor... | Yes |
Corollary 6.3.9. Let \( G \) be \( \mathbf{SL}\left( {n,\mathbb{C}}\right) \) for \( n \geq 2,\operatorname{Spin}\left( {n,\mathbb{C}}\right) \) for \( n \geq 3 \), or \( \mathbf{Sp}\left( {n,\mathbb{C}}\right) \) for \( n \geq 1 \) . Then \( G \) is algebraically simply connected. Furthermore, for every \( \lambda \in... | Proof. We know that \( G \) is connected by Theorems 2.2.5 and 6.3.5. Let \( \mathfrak{g} = \operatorname{Lie}\left( G\right) \) . Then, by Theorem 3.3.12, every finite-dimensional representation of \( \mathfrak{g} \) is completely reducible. Hence every finite-dimensional representation of \( \mathfrak{g} \) is the di... | Yes |
Lemma 6.4.1. Let \( {V}_{0} \) be an \( n \) -dimensional real subspace of \( V \) such that \( {V}_{0} \cap \mathrm{i}{V}_{0} = \) \( \{ 0\} \) . Then \( {V}_{0} \) is a real form of \( V \) . | Proof. Since \( {\dim }_{\mathbb{R}}V = {2n} \) and \( {V}_{0} \cap \mathrm{i}{V}_{0} = \{ 0\} \), one has \( {\dim }_{\mathbb{R}}\left( {{V}_{0} + \mathrm{i}{V}_{0}}\right) = {2n} \) . Thus \( V = {V}_{0} \oplus \mathrm{i}{V}_{0} \) | Yes |
Lemma 6.4.2. Let \( \mathcal{A} \) be an \( n \) -dimensional algebra over \( \mathbb{C} \) . Suppose that \( {e}_{1},\ldots ,{e}_{n} \) is a basis of \( \mathcal{A} \) over \( \mathbb{C} \) such that\n\n\[ \n{e}_{i}{e}_{j} = \mathop{\sum }\limits_{k}{a}_{ij}^{k}{e}_{k}\;\text{ with }{a}_{ij}^{k} \in \mathbb{R}\text{ f... | This is just a direct reformulation in terms of bases of the definition of real form of an algebra over \( \mathbb{C} \) . | Yes |
Theorem 6.4.3. If \( {\beta }_{0} \) is negative definite on \( {V}_{0} \), then the groups \( \operatorname{Pin}\left( {{V}_{0},{\beta }_{0}}\right) \) and \( \operatorname{\mathbf{S} \mathbf{p} \mathbf{i} \mathbf{n} }\left( {{V}_{0},{\beta }_{0}}\right) \) are compact real forms of \( \operatorname{\mathbf{P} \mathbf... | Proof. Let \( \left\{ {{e}_{1},\ldots ,{e}_{n}}\right\} \) be a basis of \( {V}_{0} \) such that \( \beta \left( {{e}_{i},{e}_{j}}\right) = - {\delta }_{ij} \) . In this basis \( {V}_{0} \) is identified with \( {\mathbb{R}}^{n} \) and \( {\beta }_{0}\left( {x, y}\right) = - \left( {{x}_{1}{y}_{1} + \cdots + {x}_{n}{y}... | Yes |
Corollary 7.1.2. Let \( h = \operatorname{diag}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) . Then\n\n\[{\Delta }_{n}\left( h\right) \operatorname{tr}\left( {{\pi }_{n}^{\mu }\left( h\right) }\right) = \mathop{\sum }\limits_{{s \in {\mathfrak{S}}_{n}}}\operatorname{sgn}\left( s\right) {h}^{s\left( {\mu + {\rh... | Proof. Write \( h = z{h}_{0} \), where \( z \in {\mathbb{C}}^{ \times } \) and \( \det \left( {h}_{0}\right) = 1 \) . Then\n\n\[ \operatorname{tr}\left( {{\pi }_{n}^{\mu }\left( h\right) }\right) = {z}^{{m}_{1} + \cdots + {m}_{n}}\operatorname{tr}\left( {{\pi }_{n}^{\mu }\left( {h}_{0}\right) }\right) .\]\n\nHere the W... | Yes |
Corollary 7.1.3 (Weyl Denominator Formula). The Weyl function is the skew-symmetrization of the character \( {\mathrm{e}}^{\rho } \) of \( H \) :\n\n\[ \n{\Delta }_{G} = \mathop{\sum }\limits_{{s \in W}}\operatorname{sgn}\left( s\right) {\mathrm{e}}^{s \cdot \rho }. \n\] | Proof. Take \( \lambda = 0 \) in the Weyl Character Formula; then \( \operatorname{ch}{V}^{0} = 1 \) . | No |
Corollary 7.1.4. Suppose \( \mu \in P \) . If \( \mu + \rho = t \cdot \left( {\lambda + \rho }\right) \) for some \( t \in W \), then\n\n\[ \mathop{\sum }\limits_{{s \in W}}\operatorname{sgn}\left( s\right) {m}_{\lambda }\left( {\mu + \rho - s \cdot \rho }\right) = \operatorname{sgn}\left( t\right) . \]\n\n(7.6)\n\nOth... | Proof. Expressing \( {\Delta }_{G} \) as an alternating sum over \( W \) by the Weyl denominator formula, we can write (7.2) as\n\n\[ \mathop{\sum }\limits_{{s \in W}}\left\{ {\mathop{\sum }\limits_{{\mu \in P}}\operatorname{sgn}\left( s\right) {m}_{\lambda }\left( \mu \right) {\mathrm{e}}^{\mu + s \cdot \rho }}\right\... | Yes |
Corollary 7.1.6. The outer multiplicity of \( {V}^{\lambda } \) is the skew-symmetrization over \( s \in W \) of the multiplicities of the weights \( \lambda + \rho - s \cdot \rho \) : | Proof. For any weight \( v \in P \), the weight space \( F\left( v\right) \) has dimension\n\n\[ \dim F\left( v\right) = \mathop{\sum }\limits_{{\mu \in {P}_{+ + }}}{\operatorname{mult}}_{F}\left( {V}^{\mu }\right) {m}_{\mu }\left( v\right) . \]\n\nTake \( v = \lambda + \rho - s \cdot \rho \) with \( s \in W \) in this... | Yes |
Corollary 7.1.7. Let \( \mu, v \in {P}_{+ + } \) . The tensor product \( {V}^{\mu } \otimes {V}^{v} \) decomposes with multiplicities\n\n\[ \n{\operatorname{mult}}_{{V}^{\mu } \otimes {V}^{\nu }}\left( {V}^{\lambda }\right) = \mathop{\sum }\limits_{{t \in W}}\operatorname{sgn}\left( t\right) {m}_{\mu }\left( {\lambda +... | Proof. Set \( F = {V}^{\mu } \otimes {V}^{\lambda } \) . By (5.59) we have\n\n\[ \n\dim F\left( \gamma \right) = \mathop{\sum }\limits_{{\alpha \in P}}{m}_{\mu }\left( \alpha \right) {m}_{\nu }\left( {\gamma - \alpha }\right)\n\]\n\nfor all \( \gamma \in P \) . Substituting this into (7.7), we obtain\n\n\[ \n{\operator... | Yes |
Lemma 7.1.8. If \( f \in \mathcal{A}\left\lbrack {\mathfrak{h}}^{ * }\right\rbrack \), then\n\n\[ \varepsilon \left( {D\left( {f{\Delta }_{G}}\right) }\right) = \varepsilon \left( {{fD}\left( {\Delta }_{G}\right) }\right) . \] | Proof. For every subset \( Q \) of \( {\Phi }^{ + } \) we define\n\n\[ {F}_{Q} = \mathop{\prod }\limits_{{\beta \in Q}}\left( {{\mathrm{e}}^{\beta /2} - {\mathrm{e}}^{-\beta /2}}\right) ,\;{\partial }_{Q} = \mathop{\prod }\limits_{{\alpha \in Q}}{\partial }_{\alpha }. \]\n\nThen \( {\Delta }_{G} = {F}_{{\Phi }^{ + }} \... | Yes |
Theorem 7.1.9 (Weyl Dimension Formula). The dimension of \( {V}^{\lambda } \) is a polynomial of degree \( \left| {\Phi }^{ + }\right| \) in \( \lambda \) : | Proof. From (7.14) and (7.15) with \( f = \operatorname{ch}\left( {V}^{\lambda }\right) \), we have\n\n\[ \varepsilon \left( {\operatorname{ch}\left( {V}^{\lambda }\right) D\left( {\Delta }_{G}\right) }\right) = \left| W\right| \mathop{\prod }\limits_{{\alpha \in {\Phi }^{ + }}}\left( {\lambda + \rho ,\alpha }\right) .... | Yes |
Theorem 7.1.10. For \( \lambda \in {P}_{+ + } \) and \( b \in \mathcal{B} \) one has\n\n\[ \n{\chi }_{\lambda }\left( b\right) = \text{ coefficient of }{x}^{\lambda + \rho }\text{ in }{\Delta }_{G}\left( x\right) {\operatorname{tr}}_{F}\left( {\pi \left( x\right) b}\right) \n\]\n\n(7.20)\n\n(where \( x \in H \) ). | Proof. We note from (7.19) that\n\n\[ \n{\operatorname{tr}}_{F}\left( {\pi \left( g\right) b}\right) = \mathop{\sum }\limits_{{\lambda \in \operatorname{Spec}\left( \pi \right) }}{\varphi }_{\lambda }\left( g\right) {\chi }_{\lambda }\left( b\right) \;\text{ for }g \in G\text{ and }b \in \mathcal{B}.\n\]\n\n(7.21)\n\nB... | Yes |
Theorem 7.1.11. For \( \lambda \in {P}_{+ + } \) and \( b \in \mathcal{B} \) one has\n\n\[ \n{\chi }_{\lambda }\left( b\right) = \mathop{\sum }\limits_{{s \in W}}\operatorname{sgn}\left( s\right) {\operatorname{tr}}_{F\left( {\lambda + \rho - s \cdot \rho }\right) }\left( b\right) .\n\]\n\n(7.23)\n\nIn particular,\n\n\... | Proof. For \( \zeta \in \mathbb{C} \) the generalized \( \zeta \) -eigenspace\n\n\[ \n{F}_{\zeta } = \left\{ {v \in F : {\left( b - \zeta \right) }^{k}v = 0\text{ for some }k}\right\}\n\]\n\nis invariant under \( G \), and we have \( F = {\bigoplus }_{\zeta \in \mathbb{C}}{F}_{\zeta } \) . Therefore, replacing \( F \) ... | No |
Lemma 7.2.1. The functions \( \left\{ {{A}_{\mu } : \mu \in {P}_{+ + }^{\mathrm{{reg}}}}\right\} \) give a basis for \( \mathcal{A}{\left\lbrack P\right\rbrack }_{\text{skew }} \) . | Proof. By (7.25) each nonzero elementary skew-symmetric function is determined (up to sign) by a Weyl group orbit in \( {P}^{\text{reg }} \), and we know from Proposition 3.1.20 that each such orbit contains a unique dominant weight. Hence if \( \mu \) and \( v \) are distinct dominant regular weights, then the orbits ... | Yes |
Proposition 7.2.2. The Weyl function \( {\Delta }_{\mathfrak{g}} \) equals \( {A}_{\rho } \) . | Proof. We can expand the product defining \( {\Delta }_{\mathfrak{g}} \) to obtain the formula\n\n\[ \n{\Delta }_{\mathfrak{g}} = {\mathrm{e}}^{\rho } + \mathop{\sum }\limits_{{\varnothing \neq M \subset {\Phi }^{ + }}}{\varepsilon }_{M}{\mathrm{e}}^{\rho -\langle M\rangle }, \n\]\n\n(7.26)\n\nwhere \( {\varepsilon }_{... | Yes |
Lemma 7.2.3. There exist integers \( {n}_{\gamma } \) such that\n\n\[ \n{\Delta }_{\mathfrak{g}}{\chi }_{\lambda } = {A}_{\lambda + \rho } + \mathop{\sum }\limits_{\gamma }{n}_{\gamma }{A}_{\gamma }\n\]\n\n(7.30)\n\nwith the sum over \( \gamma \in {P}_{+ + }^{\text{reg }} \) such that \( \gamma \prec \lambda + \rho \) ... | Proof. By Corollary 3.2.3 we have\n\n\[ \n{\chi }_{\lambda } = {\mathrm{e}}^{\lambda } + \mathop{\sum }\limits_{{\mu \prec \lambda }}{m}_{\lambda }\left( \mu \right) {\mathrm{e}}^{\mu }.\n\]\n\nHence by Proposition 7.2.2 we can write \( {\Delta }_{\mathfrak{g}}{\chi }_{\lambda } = {A}_{\lambda + \rho } + B \), where\n\... | Yes |
Proposition 7.2.6. For \( T \in \operatorname{End}\left( {V}^{\lambda }\right) \) define \( {f}_{T} \in \mathcal{O}\left\lbrack G\right\rbrack \) by \( {f}_{T}\left( g\right) = \operatorname{tr}\left( {\pi \left( g\right) T}\right) \) . Then\n\n\[ \mathrm{d}R\left( C\right) {f}_{T} = \left( {\left( {\lambda + \rho ,\la... | Proof. By (1.45) we have \( \mathrm{d}R\left( A\right) {f}_{T} = {X}_{A}{f}_{T} = {f}_{\mathrm{d}{\pi }^{\lambda }\left( A\right) T} \) for all \( A \in \mathfrak{g} \) . Extending \( \mathrm{d}R \) and \( \mathrm{d}{\pi }^{\lambda } \) to representations of \( U\left( \mathfrak{g}\right) \), we obtain \( \mathrm{d}R\l... | Yes |
Theorem 7.2.7. Let \( \varphi \) be an invariant regular function on \( G \) . Then, for \( t \in H \) ,\n\n\[ \n{\Delta }_{G}\left( t\right) \left( {\mathrm{d}R\left( C\right) \varphi }\right) \left( t\right) = \left( {\mathop{\sum }\limits_{{i = 1}}^{l}{X}_{{h}_{i}}^{2} - \left( {\rho ,\rho }\right) }\right) {\Delta ... | We shall prove Theorem 7.2.7 by reducing the calculation to the case \( G = \) \( \mathrm{{SL}}\left( {2,\mathbb{C}}\right) \) . Let\n\n\[ \ne = \left\lbrack \begin{array}{ll} 0 & 1 \\ 0 & 0 \end{array}\right\rbrack ,\;f = \left\lbrack \begin{array}{ll} 0 & 0 \\ 1 & 0 \end{array}\right\rbrack ,\;h = \left\lbrack \begin... | Yes |
Lemma 7.2.9. For each \( \alpha \in {\Phi }^{ + } \) there is an algebraic group homomorphism \( \psi : \mathbf{{SL}}\left( {2,\mathbb{C}}\right) \rightarrow G \) such that \( \mathrm{d}\psi \left( e\right) = {e}_{\alpha },\mathrm{\;d}\psi \left( f\right) = {e}_{-\alpha } \), and \( \mathrm{d}\psi \left( h\right) = {h}... | Proof. We may assume that \( G \) is an algebraic subgroup of \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \), so that we have \( \mathfrak{g} \subset \mathfrak{{gl}}\left( {n,\mathbb{C}}\right) \) . Hence there is a Lie algebra representation \( \pi \) of \( \mathfrak{{sl}}\left( {2,\mathbb{C}}\right) \) on \( {\mathb... | Yes |
For \( \lambda \in {P}_{+ + }^{\mathrm{{reg}}} \) the invariant function \( {S}_{\lambda } \) on \( G \) is an eigenfunction of the differential operator \( \mathrm{d}R\left( C\right) \): \[ \mathrm{d}R\left( C\right) {S}_{\lambda } = \left( {\left( {\lambda ,\lambda }\right) - \left( {\rho ,\rho }\right) }\right) {S}_... | Proof. Since \( \mathrm{d}R\left( C\right) \) commutes with left and right translations by elements of \( G \), the functions on both sides of equation (7.41) are \( G \) -invariant. Hence it is enough to verify this equation on the set \( {H}^{\prime } \) . By (7.33) we have \[ \mathrm{d}R\left( C\right) {S}_{\lambda ... | Yes |
Proposition 7.3.2. Let \( T = \exp \mathrm{t} \) . Then the following hold:\n\n1. \( T = H \cap U \) and \( T \) is a closed maximal commutative subgroup of \( U \) .\n\n2. \( T \) is a compact torus in \( U \) of rank \( l = \dim H \) . | Proof. (1): Since \( H = \exp \mathfrak{h} \), the polar decomposition of \( G \) (Theorem 11.5.9) shows that \( H \cap U = T \) . If \( k \in U \) commutes with \( T \), then \( \operatorname{Ad}\left( k\right) h = h \) for all \( h \in \mathfrak{t} \) . But \( \mathfrak{h} = \mathfrak{t} + \mathrm{{it}} \), so we hav... | Yes |
Lemma 7.3.4. Fix \( g \in U \) and \( h \in T \) and define the map\n\n\[ \psi = {L}_{g{h}^{-1}{g}^{-1}} \circ \varphi \circ {L}_{\left( g, h\right) } : \left( {U/T}\right) \times T \rightarrow U. \]\n\nThen \( \psi \left( {o, I}\right) = I \) and\n\n\[ \mathrm{d}{\psi }_{\left( o, I\right) }\left( {X \oplus Y}\right) ... | Proof. By definition,\n\n\[ {\left. \mathrm{d}{\psi }_{\left( o, I\right) }\left( X \oplus Y\right) = \frac{d}{dt}\left\{ g{h}^{-1}\exp \left( tX\right) h\exp \left( tY\right) \exp \left( -tX\right) {g}^{-1}\right\} \right| }_{t = 0} \]\n\n\[ = {\left. \frac{d}{dt}\left\{ g\exp \left( t\operatorname{Ad}\left( {h}^{-1}\... | Yes |
Theorem 7.3.6. The restriction of \( \varphi \) to \( \left( {U/T}\right) \times {T}^{\prime \prime } \) is a covering map of degree \( \left| {W}_{G}\right| \) . Furthermore, \[ {\left( {\varphi }^{ * }{\omega }_{U}\right) }_{u \cdot o, h} = {\left| {\Delta }_{G}\left( h\right) \right| }^{2}{\left( {\omega }_{T}\right... | Proof. From Lemma 7.3.4 we see that \( \mathrm{d}{\varphi }_{\left( u \cdot o, h\right) } \) is nonsingular for all \( u \in U \) and \( h \in {T}^{\prime \prime } \) . Since \( \dim \left( {U/T}\right) + \dim \left( T\right) = \dim U \), we conclude from the inverse function theorem that the restriction of \( \varphi ... | Yes |
Corollary 7.3.7 (Weyl Integral Formula). Let \( f \in C\left( U\right) \) . Then\n\n\[ \n{\int }_{U}f\left( u\right) \mathrm{d}u = \frac{1}{\left| {W}_{G}\right| }{\int }_{T}{\left| {\Delta }_{G}\left( h\right) \right| }^{2}\left\{ {{\int }_{U/T}f\left( {{uh}{u}^{-1}}\right) \mathrm{d}\dot{u}}\right\} \mathrm{d}h, \]\n... | Proof. The complement of \( {T}^{\prime \prime } \) has measure zero in \( T \) . Hence we may replace the integral over \( T \) by the integral over \( {T}^{\prime \prime } \) on the right side of (7.49) without changing the value of the integral. With this done, formula (7.49) follows immediately from Theorems 7.3.6 ... | No |
Corollary 7.3.8. The map \( \varphi : \left( {U/T}\right) \times T \rightarrow U \) is surjective. Hence every element of \( U \) is \( U \) -conjugate to an element of \( T \) . | Proof. Since \( \left( {U/T}\right) \times T \) is compact, the image of \( \varphi \) is closed in \( U \) . If it were not all of \( U \), there would be a nonzero function \( f \in C\left( U\right) \) such that \( f \geq 0 \) and \( f = 0 \) on the image of \( U \) . However, then \( f\left( {{uh}{u}^{-1}}\right) = ... | Yes |
Lemma 7.3.9. Suppose \( \lambda ,\mu \in {P}_{+ + }^{\text{reg }} \) . Then\n\n\[ \frac{1}{\left| W\right| }{\int }_{T}{A}_{\lambda }\left( t\right) \overline{{A}_{\mu }\left( t\right) }\mathrm{d}t = \left\{ \begin{array}{l} 0\text{ if }\lambda \neq \mu , \\ 1\text{ if }\lambda = \mu . \end{array}\right. \] | Proof. If \( \lambda \neq \mu \) then the orbits \( \{ s \cdot \lambda : s \in W\} \) and \( \{ s \cdot \mu : s \in W\} \) are disjoint and have \( \left| W\right| \) elements, since \( \lambda ,\mu \) are dominant and regular (see Proposition 3.1.20). Hence for \( s,{s}^{\prime } \in W \) we have\n\n\[ {\int }_{T}{t}^... | Yes |
Proposition 7.3.10. Suppose \( \varphi \in \mathcal{O}\left\lbrack H\right\rbrack \) satisfies \( \varphi \left( {{sh}{s}^{-1}}\right) = \operatorname{sgn}\left( s\right) \varphi \left( h\right) \) for all \( s \in \) W. Then\n\n\[ \varphi = \mathop{\sum }\limits_{{\mu \in {P}_{+ + }^{\text{reg }}}}c\left( \mu \right) ... | Proof. The function \( \varphi \) has an expansion\n\n\[ \varphi \left( h\right) = \mathop{\sum }\limits_{{\mu \in P}}c\left( \mu \right) {h}^{\mu } \]\n\nwith \( c\left( \mu \right) \in \mathbb{C} \) and \( c\left( \mu \right) = 0 \) for all but a finite number of \( \mu \) . Since this expansion is unique, the skew s... | Yes |
Theorem 8.1.4. The branching from \( \operatorname{\mathbf{S} \mathbf{p} \mathbf{i} \mathbf{n} }\left( {2n}\right) \) to \( \operatorname{\mathbf{S} \mathbf{p} \mathbf{i} \mathbf{n} }\left( {{2n} - 1}\right) \) is multiplicity-free. The multiplicity \( m\left( {\lambda ,\mu }\right) \) is 1 if and only if \( \lambda \)... | \[ {\lambda }_{1} \geq {\mu }_{1} \geq {\lambda }_{2} \geq \cdots \geq {\lambda }_{n - 1} \geq {\mu }_{n - 1} \geq \left| {\lambda }_{n}\right| \] | No |
Proposition 8.1.6. Let \( \lambda \in {\mathbb{N}}_{+ + }^{n} \) and let \( \gamma = \left\{ {{\mu }^{\left( n\right) },\ldots ,{\mu }^{\left( 1\right) }}\right\} \) be an \( n \) -fold branching pattern of shape \( \lambda \) . There is a unique flag of subspaces \( {F}_{n}^{\lambda } \supset {M}_{n - 1}^{\gamma } \su... | Proof. This follows from Theorem 8.1.2 by induction on \( n \) . | No |
Corollary 8.1.7 (Gelfand-Cetlin Basis). Let \( \lambda \in {\mathbb{N}}_{+ + }^{n} \) . The set \( \left\{ {u}_{\gamma }\right\} \), where \( \gamma \) ranges over all \( n \) -fold branching patterns of shape \( \lambda \), is a basis for \( {F}_{n}^{\lambda } \) . Hence the weights of \( {F}_{n}^{\lambda } \) are in ... | Proof. This follows from Theorem 8.1.2 by induction on \( n \) . | No |
Theorem 8.2.1 (Branching Multiplicity Formula). Assume that the pair \( \mathfrak{g},\mathfrak{h} \) satisfies condition \( \left( \mathrm{R}\right) \) . Then the branching multiplicities are\n\n\[ m\left( {\lambda ,\mu }\right) = \mathop{\sum }\limits_{{s \in {W}_{\mathfrak{g}}}}\operatorname{sgn}\left( s\right) {\wp ... | Proof. Let \( {\varphi }_{\lambda } \) be the character of the irreducible \( \mathfrak{g} \) -module with highest weight \( \lambda \) and let \( {\psi }_{\mu } \) be the character of the irreducible \( \mathfrak{h} \) -module with highest weight \( \mu \) . If \( \overline{{\varphi }_{\lambda }} \) is the restriction... | Yes |
Corollary 8.2.2 (Weight Multiplicity Formula). The multiplicity of the weight \( \mu \) is the alternating sum | \[ {m}_{\lambda }\left( \mu \right) = \mathop{\sum }\limits_{{s \in W}}\operatorname{sgn}\left( s\right) \wp \left( {s \cdot \left( {\lambda + \rho }\right) - \mu - \rho }\right) . \] | Yes |
Lemma 8.3.1. Let \( \lambda ,\mu \in {\mathbb{Z}}_{+ + }^{n} \). Then \( \det {A}_{n}\left( {\lambda ,\mu }\right) = 1 \) if\n\n\[{\lambda }_{1} \geq {\mu }_{1} \geq {\lambda }_{2} \geq \cdots \geq {\lambda }_{n - 1} \geq {\mu }_{n - 1} \geq {\lambda }_{n} \geq {\mu }_{n}.\]\n\nOtherwise, \( \det {A}_{n}\left( {\lambda... | Proof. We proceed by induction on \( n \), the case \( n = 1 \) being the definition of the function \( a \mapsto {\left\lbrack a\right\rbrack }_{ + } \). Assume that the lemma is true for \( n - 1 \) and take \( \lambda ,\mu \in {\mathbb{Z}}_{+ + }^{n} \). If \( {\mu }_{1} > {\lambda }_{1} \) then \( {\mu }_{1} > {\la... | Yes |
Lemma 9.1.1. Let \( \lambda \) be a partition of \( k \) with \( p \) parts. Then for all \( n \geq p \) , \[ {\left( {\bigotimes }^{k}{\mathbb{C}}^{n}\right) }^{{N}_{n}^{ + }}\left( \lambda \right) = {\left( {\bigotimes }^{k}{\mathbb{C}}^{p}\right) }^{{N}_{p}^{ + }}\left( \lambda \right) . \] | Proof. Let \( i > p \) . From (9.3) we have \( {\rho }_{k}\left( {e}_{i, i + 1}\right) u = 0 \) for all \( u \in \left( {{\bigotimes }^{k}{\mathbb{C}}^{n}}\right) \left( \lambda \right) \), where \( {e}_{ij} \in {M}_{n}\left( \mathbb{C}\right) \) are the usual elementary matrices. Since \( {\mathfrak{n}}_{n}^{ + } \) i... | Yes |
Proposition 9.1.3. Let \( \mu \in \operatorname{Par}\left( {k, n}\right) \) . Then the induced module \( {I}^{\mu } \) for \( {\mathfrak{S}}_{k} \) decomposes as\n\n\[ \n{I}^{\mu } \cong {\bigoplus }_{\lambda \in \operatorname{Par}\left( {k, n}\right) }{K}_{\lambda \mu }{G}^{\lambda }.\n\]\n\nIn particular, \( {G}^{\mu... | Proof. By Proposition 4.4.6 and Theorem 9.1.2 we have\n\n\[ \n{I}^{\mu } \cong {\bigoplus }_{\lambda \in \operatorname{Par}\left( {k, n}\right) }{F}_{n}^{\lambda }\left( \mu \right) \otimes {G}^{\lambda }\n\]\n\nas a module for \( {\mathfrak{S}}_{k} \) . Since \( {K}_{\mu \lambda } = \dim {F}_{n}^{\lambda }\left( \mu \... | Yes |
Theorem 9.1.4. Let \( y \in {\mathfrak{S}}_{k} \) . Then\n\n\[ \n{\operatorname{ch}}_{{\mathfrak{S}}_{k}}\left( {G}^{\lambda }\right) \left( y\right) = \mathop{\sum }\limits_{{s \in {\mathfrak{S}}_{n}}}\operatorname{sgn}\left( s\right) \# \left\{ {\text{ fixed points of }y\text{ on }{\mathfrak{S}}_{k}/{\mathfrak{S}}_{\... | Proof. By Theorem 7.1.11 and equation (9.5), we have\n\n\[ \n{\operatorname{ch}}_{{\mathfrak{S}}_{k}}\left( {G}^{\lambda }\right) = \mathop{\sum }\limits_{{s \in {\mathfrak{S}}_{n}}}\operatorname{sgn}\left( s\right) {\operatorname{ch}}_{{\mathfrak{S}}_{k}}\left( {\left( {{ \otimes }^{k}{\mathbb{C}}^{n}}\right) \left( {... | Yes |
Corollary 9.1.5. Let \( \lambda \in \operatorname{Par}\left( {k, n}\right) \) . Set \( \mu = \lambda + \rho \) . Then\n\n\[ \dim {G}^{\lambda } = \frac{k!}{{\mu }_{1}!{\mu }_{2}!\cdots {\mu }_{n}!}\mathop{\prod }\limits_{{1 \leq i < j \leq n}}\left( {{\mu }_{i} - {\mu }_{j}}\right) . \] | Proof. For \( \gamma \in {\mathbb{N}}^{n} \) with \( \left| \gamma \right| = k \), the cardinality of \( {\mathfrak{S}}_{k}/{\mathfrak{S}}_{\gamma } \) is the multinomial coefficient \( \left( \begin{array}{l} k \\ \gamma \end{array}\right) = k!/\prod {\gamma }_{i}! \) . Hence taking \( y = 1 \) in (9.7) gives\n\n\[ \d... | Yes |
Corollary 9.1.6 (Hook-Length Formula). Let \( \lambda \in \operatorname{Par}\left( k\right) \) . Then\n\n\[ \dim {G}^{\lambda } = \frac{k!}{\mathop{\prod }\limits_{{\left( {i, j}\right) \in \lambda }}{h}_{ij}}. \] | Proof. We use induction on the number of columns of the Ferrers diagram of \( \lambda \) . If the diagram has one column, then \( \dim {G}^{\lambda } = 1 \), since \( \lambda \) corresponds to the sgn representation. In this case \( {h}_{i1} = k + 1 - i \), so that the product of the hook lengths is \( k \) !, and henc... | Yes |
Theorem 9.1.7 (Frobenius Character Formula). Let \( \lambda = \left\lbrack {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right\rbrack \) be a partition of \( k \) with at most \( n \) parts. Set \[ \mu = \lambda + \rho = \left\lbrack {{\lambda }_{1} + n - 1,{\lambda }_{2} + n - 2,\ldots ,{\lambda }_{n}}\right\rbrack . \] The... | Proof. This follows immediately from (9.13) and Theorem 7.1.10. | No |
Proposition 9.2.1. Let \( \mathcal{A} \) (respectively, \( {\mathcal{A}}^{\prime } \) ) be the subalgebra of \( \operatorname{End}\left( Y\right) \) generated by \( \rho \left( K\right) \) (respectively, \( \rho \left( {K}^{\prime }\right) \) ). The following are equivalent:\n\n1. All multiplicities \( {m}_{\pi ,{\pi }... | Proof. The implication (2) \( \Rightarrow \) (1) follows directly from Theorem 4.2.1. Now assume that (1) holds and suppose \( {m}_{\pi ,{\pi }^{\prime }} = 1 \) for some pair \( \left( {\pi ,{\pi }^{\prime }}\right) \) . The \( {\pi }^{\prime } \) -isotypic subspace of \( Y \) (viewed as a \( {K}^{\prime } \) -module)... | Yes |
Theorem 9.2.2 (Seesaw Reciprocity). Let \( K \times {K}^{\prime } \) and \( G \times {G}^{\prime } \) be a seesaw pair; relative to a \( \left( {G \times {K}^{\prime }}\right) \) -module \( Y \) . Let \( \pi \in {\widehat{K}}_{Y} \) and \( {\pi }^{\prime } \in {\widehat{K}}_{Y}^{\prime } \) determine the same isotypic ... | Proof. Let \( \sigma \in \widehat{G} \) . Then the isotypic decomposition of \( {V}_{\sigma } \) as a \( K \) -module is\n\n\[ \n{V}_{\sigma } \cong {\bigoplus }_{\tau \in \widehat{K}}{\operatorname{Hom}}_{K}\left( {\tau ,{\operatorname{Res}}_{K}^{G}\left( \sigma \right) }\right) \otimes {V}_{\tau },\n\]\n\nwith \( k \... | Yes |
Theorem 9.2.3. Let \( \lambda ,\mu, v \) be Ferrers diagrams and \( k, m, n \) positive integers with \( \operatorname{depth}\left( \lambda \right) \leq n,\operatorname{depth}\left( \mu \right) \leq \min \{ k, n\} \), and \( \operatorname{depth}\left( \nu \right) \leq \min \{ m, n\} \) . Then 9.2 Dual Reductive Pairs\n... | Proof. Let \( K = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) ,{K}^{\prime } = \mathbf{{GL}}\left( {k + m,\mathbb{C}}\right), G = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \times \mathbf{{GL}}\left( {n,\mathbb{C}}\right) ,{G}^{\prime } = \) \( \mathbf{{GL}}\left( {k,\mathbb{C}}\right) \times \mathbf{{GL}}\left( {m,\mathb... | Yes |
Corollary 9.2.4 (Pieri’s Rule). Let \( \mu \) be a diagram of depth \( \leq n - 1 \) and \( \nu \) a diagram of depth one. Then\n\n\[ \n{\pi }_{n}^{\mu } \otimes {\pi }_{n}^{\nu } \cong {\bigoplus }_{\lambda }{\pi }_{n}^{\lambda }\n\]\n\nwhere the sum is over all diagrams \( \lambda \) of depth \( \leq n \) such that \... | Proof. By the branching law from \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) to \( \mathbf{{GL}}\left( {n - 1,\mathbb{C}}\right) \times \mathbf{{GL}}\left( {1,\mathbb{C}}\right) \) (Theorem 8.1.2), we have\n\n\[ \n{d}_{\mu \nu }^{\lambda } = \left\{ \begin{array}{ll} 1 & \text{ if }\mu \text{ interlaces }\lambda \t... | Yes |
Theorem 9.2.5. Let \( \lambda \in \operatorname{Par}\left( {k + m, n}\right) \) . Then\n\n\[ \n{\operatorname{Res}}_{{\mathfrak{S}}_{k} \times {\mathfrak{S}}_{m}}^{{\mathfrak{S}}_{k + m}}\left( {\sigma }^{\lambda }\right) \cong {\bigoplus }_{\mu, v}{c}_{\mu v}^{\lambda }{\sigma }^{\mu }\widehat{ \otimes }{\sigma }^{v},... | Proof. Assume that \( \lambda \) has depth \( n \) . We take the seesaw pairs\n\n\[ \nK = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \subset G = \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \times \mathbf{{GL}}\left( {n,\mathbb{C}}\right) ,\n\]\n\n\[ \n{G}^{\prime } = {\mathfrak{S}}_{k} \times {\mathfrak{S}}_{m} \subset {K... | Yes |
Corollary 9.2.6. Let \( \mu \in \operatorname{Par}\left( {k, n}\right) \) and \( v \in \operatorname{Par}\left( {m, n}\right) \) . Then\n\n\[{\operatorname{Ind}}_{{\mathfrak{S}}_{k} \times {\mathfrak{S}}_{m}}^{{\mathfrak{S}}_{k + m}}\left( {{\sigma }^{\mu }\widehat{ \otimes }{\sigma }^{\nu }}\right) = {\bigoplus }_{\la... | Proof. Use Theorem 9.2.5 and Frobenius reciprocity (Theorem 4.4.1). | No |
Corollary 9.2.7 (Branching Rule). Let \( \\lambda \\in \\operatorname{Par}\\left( n\\right) \) . Then\n\n\[ \n{\\operatorname{Res}}_{{\\mathfrak{S}}_{n - 1}}^{{\\mathfrak{S}}_{n}}\\left( {\\sigma }^{\\lambda }\\right) \\cong {\\bigoplus }_{\\mu }{\\sigma }^{\\mu }\n\]\n\nwith the sum over all \( \\mu \\in \\operatornam... | Proof. Take \( k = n - 1 \) and \( m = 1 \) in Theorem 9.2.5 and use the calculation of the Littlewood-Richardson coefficients in the proof of Corollary 9.2.4. | No |
Theorem 9.2.8. Let \( X = {M}_{k, n}\left( \mathbb{C}\right) \) . The \( {\det }_{k} \)-weight space in \( \mathcal{P}\left( X\right) \) decomposes under the action of \( {\mathfrak{S}}_{k} \times \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) as\n\n\[ \mathcal{P}\left( X\right) \left( {\det }_{k}\right) \cong {\bigoplus... | Proof. We have \( \mathcal{P}\left( X\right) \left( {\det }_{k}\right) \subset {\mathcal{P}}^{k}\left( X\right) \). Hence Theorem 5.6.7 implies the decomposition (9.28). The irreducibility of \( {F}_{k}^{\lambda }\left( {\det }_{k}\right) \) under \( {\mathfrak{S}}_{k} \) and its equivalence to \( {\sigma }^{\lambda } ... | Yes |
Lemma 9.3.2. Let \( \lambda \in \operatorname{Par}\left( {k, n}\right) \) . If \( A \) has shape \( \lambda \) then \( \mathbf{c}\left( A\right) {e}_{A} \) is nonzero and \( {N}_{n}^{ + } \) - fixed of weight \( \lambda \) . | Proof. Suppose first that \( A = A\left( \lambda \right) \) . Let \( {\lambda }^{t} \) be the transposed shape (dual partition) to \( \lambda \) . Then\n\n\[ {e}_{A} = {e}_{1} \otimes {e}_{2} \otimes \cdots \otimes {e}_{{\lambda }_{1}^{t}} \otimes {e}_{1} \otimes {e}_{2} \otimes \cdots \otimes {e}_{{\lambda }_{2}^{t}} ... | Yes |
Lemma 9.3.3. Let \( \lambda ,\mu \in \operatorname{Par}\left( k\right) \) with \( k \geq 2 \) . Let \( A \) be a tableau of shape \( \lambda \) and \( B \) a tableau of shape \( \mu \) . Suppose either\n\n(i) \( \;\lambda \overset{\text{ lex }}{ < }\mu \), or else\n\n(ii) \( \lambda = \mu \) and \( c \cdot A \neq r \cd... | Proof. First suppose \( \lambda = \mu \) and argue by contradiction. If the numbers in each row of \( B \) appear in distinct columns of \( A \), then there exists \( c \in \operatorname{Col}\left( A\right) \) such that each row of \( c \cdot A \) contains the same set of numbers as the corresponding row of \( B \) . H... | Yes |
Corollary 9.3.4. (Hypotheses of Lemma 9.3.3) There exists \( \gamma \in \operatorname{Col}\left( A\right) \cap \operatorname{Row}\left( B\right) \) with \( {\gamma }^{2} = 1 \) and \( \operatorname{sgn}\left( \gamma \right) = - 1 \) . | Proof. There exists a pair of numbers \( l, m \) in the same column of \( A \) and the same row of \( B \), by Lemma 9.3.3. Hence we can take \( \gamma \in {\mathfrak{S}}_{k} \) to be the transposition of \( l \) and \( m \) . | No |
Proposition 9.3.5. Let \( \lambda \in \operatorname{Par}\left( {k, n}\right) \) . 1. If \( A \) is a tableau of shape \( \lambda \), then \( \mathbf{c}\left( A\right) \left( {{\bigotimes }^{k}{\mathbb{C}}^{n}}\right) \left( \lambda \right) = \mathbb{C}\mathbf{c}\left( A\right) {e}_{A} \) . 2. \( {G}^{\lambda } = \matho... | Proof. (1): By (9.29) it suffices to consider the action of \( \mathbf{c}\left( A\right) \) on \( {e}_{B} \), for all \( B \in \) \( \operatorname{Tab}\left( \lambda \right) \) . If there exist \( c \in \operatorname{Col}\left( A\right) \) and \( r \in \operatorname{Row}\left( B\right) \) such that \( c \cdot A = r \cd... | Yes |
Lemma 9.3.6. Let \( \lambda ,\mu \in \operatorname{Par}\left( {k, n}\right) \). 1. If \( \mu \overset{\text{ lex }}{ > }\lambda \) then \( \mathbf{c}\left( A\right) \left( {{\bigotimes }^{k}{\mathbb{C}}^{n}}\right) \left( \mu \right) = 0 \) for all \( A \in \operatorname{Tab}\left( \lambda \right) \). 2. If \( \mu \ove... | Proof. (1): Take \( B \in \operatorname{Tab}\left( \mu \right) \) and let \( \gamma \in \operatorname{Col}\left( A\right) \cap \operatorname{Row}\left( B\right) \) be as in Corollary 9.3.4. Then \( \mathbf{c}\left( A\right) {e}_{B} = 0 \) by the same calculation as (9.32). This proves (1). (2): Take \( \gamma \in \oper... | Yes |
Proposition 9.3.7. Let \( \lambda \in \operatorname{Par}\left( {k, n}\right) \) and let \( A \) be a tableau of shape \( \lambda \) . Define \( \mathbf{s}\left( A\right) = \mathbf{c}\left( A\right) \mathbf{r}\left( A\right) \) as an element of the group algebra of \( {\mathfrak{S}}_{k} \) .\n\n1. \( \mathbf{s}\left( A\... | Proof. Suppose \( \mu > \lambda \) . Since the weight spaces of \( {H}_{n} \) are invariant under the group algebra of \( {\mathfrak{S}}_{k} \), we have\n\n\[ \mathbf{s}\left( A\right) {G}^{\mu } \subset \mathbf{s}\left( A\right) \left( {{ \otimes }^{k}{\mathbb{C}}^{n}}\right) \left( \mu \right) \subset \mathbf{c}\left... | Yes |
Lemma 9.3.8. Let \( A \) be a tableau of shape \( \lambda \in \operatorname{Par}\left( k\right) \) . Then \( \mathbf{s}\left( A\right) \in {\mathcal{B}}^{\lambda } \) and \( \mathbf{s}{\left( A\right) }^{2} = \) \( {\xi }_{\lambda }\mathbf{s}\left( A\right) \), where the scalar \( {\xi }_{\lambda } \) is nonzero and is... | Proof. By Proposition 9.3.7 (2) there exists a linear functional \( {f}_{A} \in {\mathcal{E}}_{k}^{ * } \) such that\n\n\[ \mathbf{s}\left( A\right) x = {f}_{A}\left( x\right) \mathbf{c}\left( A\right) {e}_{A}\;\text{ for }x \in {\mathcal{E}}_{k}. \]\n\n(9.34)\n\nHence \( \mathbf{s}\left( A\right) \in {\mathcal{B}}^{\l... | Yes |
Lemma 9.3.9. Let \( \lambda \in \operatorname{Par}\left( k\right) \) . Then \( {\xi }_{\lambda } = k!/\dim {G}^{\lambda } \) . | Proof. Let \( L \) be the representation of \( \mathcal{A}\left\lbrack {\mathfrak{S}}_{k}\right\rbrack \) on \( \mathcal{A}\left\lbrack {\mathfrak{S}}_{k}\right\rbrack \) (viewed as a vector space) given by left convolution. For \( A \in \operatorname{Tab}\left( \lambda \right) \) define \( {\mathbf{p}}_{A} = {\xi }_{\... | Yes |
Theorem 9.3.10. Let \( \lambda \) be a partition of \( k \) with at most \( n \) parts. If \( A \) is a tableau of shape \( \lambda \), then the operator \( {\mathbf{p}}_{A} = \left( {\dim {G}^{\lambda }/k!}\right) \mathbf{s}\left( A\right) \) projects \( { \otimes }^{k}{\mathbb{C}}^{n} \) onto an irreducible \( \mathb... | Proof. From Lemma 9.3.9 we know that \( {\mathbf{p}}_{A} \) is a projection operator, and from Schur-Weyl duality we have\n\n\[ \n{\mathbf{p}}_{A}\left( {{\bigotimes }^{k}{\mathbb{C}}^{n}}\right) \cong {\mathbf{p}}_{A}\left( {{\bigotimes }_{\mu \in \mathrm{{Par}}\left( {k, n}\right) }{F}_{n}^{\mu } \otimes {G}^{\mu }}\... | Yes |
Theorem 9.3.12. The tensors \( \left\{ {\mathbf{s}\left( A\right) {e}_{A} : A \in \operatorname{STab}\left( \lambda \right) }\right\} \) give a basis for \( {G}^{\lambda } \) . | The proof will require a combinatorial lemma. We give \( \operatorname{STab}\left( \lambda \right) \) the lexicographic order \( > \) defined by reading the entries of \( A \in \operatorname{STab}\left( \lambda \right) \) from left to right along each row from the top row to the bottom row. For example, | No |
Lemma 9.3.13. Let \( A,{A}^{\prime } \in \operatorname{STab}\left( \lambda \right) \) and suppose \( A\overset{\text{ lex }}{ > }{A}^{\prime } \) . Then \( \mathbf{s}\left( {A}^{\prime }\right) \mathbf{s}\left( A\right) = 0 \) . | Proof. We may assume that\n\n1. the first \( p - 1 \) rows of \( A \) and \( {A}^{\prime } \) are identical;\n\n2. the elements in positions \( 1,\ldots, q - 1 \) of the \( p \) th rows of \( A;{A}^{\prime } \) are identical;\n\n3. \( m = {A}_{pq} > {m}^{\prime } = {A}_{pq}^{\prime } \) .\n\nWe claim that the number \(... | Yes |
Theorem 9.3.15. Let \( \lambda \) be a partition of \( k \) with at most \( n \) parts.\n\n1. If \( U \subset { \otimes }^{k}{\mathbb{C}}^{n} \) is a subspace invariant under \( {\rho }_{k}\left( {\mathbf{{GL}}\left( {n,\mathbb{C}}\right) }\right) \), then \( {\mathbf{P}}_{\lambda }U \) is the isotypic component of \( ... | Proof. (1): Let \( \mathcal{A} = \operatorname{Span}{\rho }_{k}\left( {\mathbf{{GL}}\left( {n,\mathbb{C}}\right) }\right) \) and \( \mathcal{B} = \operatorname{Span}{\sigma }_{k}\left( {\mathfrak{S}}_{k}\right) \) . By Schur’s commutant theorem (Theorem 4.2.10) and the double commutant theorem (Theorem 4.1.13) we know ... | Yes |
Proposition 10.1.1. Let \( G \subset \mathbf{{GL}}\left( V\right) \) be the full group of isometries for a nondegenerate symmetric or skew-symmetric bilinear form \( \omega \) on \( V \) . Let \( \Gamma \subset {\mathfrak{S}}_{2k} \) be a set of representatives for the double cosets \( \tau \left( {\mathfrak{S}}_{k}\ri... | Proof. Recall from Theorem 5.3.4 that\n\n\[ \n{\operatorname{End}}_{G}\left( {{\bigotimes }^{k}V}\right) = \operatorname{Span}\left\{ {T\left( {{\sigma }_{2k}\left( s\right) {\theta }_{k}}\right) : s \in {\Xi }_{k}}\right\} ,\n\]\n\n(10.3)\n\nwhere \( {\Xi }_{k} \) is any set of representatives for the cosets \( {\math... | Yes |
Lemma 10.1.2. Suppose that \( z = \left\{ {{i}_{1},{j}_{1}}\right\} ,\ldots ,\left\{ {{i}_{r},{j}_{r}}\right\} \in {X}_{k} \) is a normalized \( r \) -bar Brauer diagram. Then\n\n\[{\tau }_{{i}_{p}{j}_{p}}{\tau }_{{i}_{q}{j}_{q}} = {\tau }_{{i}_{q}{j}_{q}}{\tau }_{{i}_{p}{j}_{p}}\;\text{ for }p \neq q.\]\n\n(10.9)\n\nT... | Proof. The commutativity relation (10.9) is clear, since \( {\tau }_{ij} \) operates on only the \( i \) th and \( j \) th tensor positions. We take \( \gamma \in {\mathfrak{S}}_{2k} \) as the product of the transpositions \( 2{i}_{p} \leftrightarrow 2{j}_{p} - 1 \) for \( p = 1,\ldots, r \) . Then (10.10) follows dire... | Yes |
Proposition 10.1.3. Let \( n = \dim V \) . The algebra \( {\mathcal{B}}_{k}\left( {\varepsilon n}\right) \) is spanned by the set of operators \( {\sigma }_{k}\left( s\right) {\tau }_{z} \) with \( s \in {\mathfrak{S}}_{k} \) and \( z \in {Z}_{k} \) . | Proof. Given \( z \in {Z}_{k} \) take \( \gamma \in {\mathfrak{S}}_{2k} \) as in Lemma 10.1.2. The set \( \Gamma \) of all such \( \gamma \) is then a set of representatives for the double cosets \( \tau \left( {\mathfrak{S}}_{k}\right) \smallsetminus {\mathfrak{S}}_{2k}/{\mathfrak{B}}_{k} \) . Now apply Proposition 10... | No |
Corollary 10.1.4. Suppose \( n \geq {2k} \) . Then the set \( \left\{ {{\sigma }_{k}\left( s\right) {\tau }_{z} : s \in {\mathfrak{S}}_{k}, z \in {Z}_{k}}\right\} \) is a basis for \( {\mathcal{B}}_{k}\left( {\varepsilon n}\right) \) . | Proof. As a vector space, \( {\mathcal{B}}_{k}\left( {\varepsilon n}\right) \) is isomorphic to \( {\left\lbrack {\bigotimes }^{2k}{V}^{ * }\right\rbrack }^{G} \), with the operator \( {\sigma }_{k}\left( s\right) {\tau }_{z} \) corresponding to the complete contraction \( {\widetilde{\lambda }}_{x} \) for \( x = s \cd... | Yes |
Lemma 10.1.5. The operators \( {\tau }_{ij} \) satisfy the following relations, where \( n = \dim V \) and (il) denotes the transposition of \( i \) and \( l \) :\n\n1. \( {\tau }_{ij}^{2} = n{\tau }_{ij} \) .\n\n2. \( {\tau }_{ij}{\tau }_{jl} = {\sigma }_{k}\left( {il}\right) {\tau }_{jl} \) for distinct \( i, j, l \)... | Proof. The contraction and expansion operators satisfy\n\n\[ \n{C}_{ij}{D}_{ij} = {nI} \n\]\n\n(10.13)\n\nwhich follows from \( \mathop{\sum }\limits_{{p = 1}}^{n}\omega \left( {{f}_{p},{f}^{p}}\right) = n \) . This implies property (1). To verify (2), note that\n\n\[ \n{\tau }_{ij}{\tau }_{jl}\left( {{v}_{1} \otimes \... | Yes |
Theorem 10.1.6. The algebra \( {\mathcal{B}}_{k}\left( {\varepsilon n}\right) \) is generated by the operators \( {\sigma }_{k}\left( s\right) \) for \( s \in {\mathfrak{S}}_{k} \) and the projection \( {P}_{k - 1} \) . | Proof. From (3) in Lemma 10.1.5 we have\n\n\[ \n{\tau }_{ij} = n{\sigma }_{k}\left( s\right) {P}_{k - 1}{\sigma }_{k}{\left( s\right) }^{-1}, \n\] \n\n(10.14) \n\nwhere \( s \in {\mathfrak{S}}_{k} \) is the product of the transpositions \( k - 1 \leftrightarrow i \) and \( k \leftrightarrow j \) . The theorem now follo... | Yes |
Theorem 10.2.1. The space \( \mathcal{H}\left( {{ \otimes }^{k}V}\right) \) is invariant under \( {\rho }_{k}\left( G\right) \) and \( {\sigma }_{k}\left( {\mathfrak{S}}_{k}\right) \) . Furthermore, the commutant of \( G \) on \( \mathcal{H}\left( {{ \otimes }^{k}V}\right) \) is \( \mathbb{C}\left\lbrack {{\sigma }_{k}... | Proof. Since \( {C}_{ij}{\tau }_{ij} = {C}_{ij}{D}_{ij}{C}_{ij} = n{C}_{ij} \), we have\n\n\[ \operatorname{Ker}\left( {C}_{ij}\right) = \operatorname{Ker}\left( {\tau }_{ij}\right) \]\n\nHence \( u \) is harmonic if and only if \( {\tau }_{ij}u = 0 \) for \( 1 \leq i < j \leq k \) . Since \( {\tau }_{ij} \) commutes w... | Yes |
Proposition 10.2.2. There are the following dichotomies:\n\n1. Assume that \( \lambda \) satisfies (10.19). Then either \( {W}^{k}\left( \lambda \right) \cap \mathcal{H}\left( {{ \otimes }^{k}{\mathbb{C}}^{n}}\right) = 0 \), or else \( {W}^{k}\left( \lambda \right) \subset \mathcal{H}\left( {{\bigotimes }^{k}{\mathbb{C... | Proof. (1): For each \( G \) -isotypic subspace \( E \) in \( { \otimes }^{k}{\mathbb{C}}^{n} \), there is a unique \( \lambda \) satisfying (10.19) that is the weight of a b-extreme tensor in \( E \) . By Theorems 4.2.1 and 4.2.12 (and using the results in Section 5.5.5 relating representations of \( \mathbf{{SO}}\lef... | Yes |
Corollary 10.2.3. Let \( \mu \) be a \( \widetilde{\mathfrak{b}} \) -dominant weight. Assume that \( \left| \mu \right| = k - {2r} \) for some integer \( r \geq 0 \), and that \( 0 \neq {W}^{k}\left( \bar{\mu }\right) \subset \mathcal{H}\left( {{\bigotimes }^{k}{\mathbb{C}}^{n}}\right) \) . Then \( r = 0 \) and \( {W}^... | Proof. Since \( \mu \) is \( \widetilde{\mathfrak{b}} \) -dominant and \( \left| \mu \right| = k - {2r} \), we have \( {\widetilde{W}}^{k - {2r}}\left( \mu \right) \neq 0 \) . Thus\n\n\[ 0 \neq {\left( {D}_{12}\right) }^{r}{\widetilde{W}}^{k - {2r}}\left( \mu \right) \subset {W}^{k}\left( \bar{\mu }\right) ,\]\n\nsince... | Yes |
Corollary 10.2.4. If \( p \neq q \) then \( {\operatorname{Hom}}_{G}\left( {\mathcal{H}\left( {{\bigotimes }^{p}{\mathbb{C}}^{n}}\right) ,\mathcal{H}\left( {{\bigotimes }^{q}{\mathbb{C}}^{n}}\right) }\right) = 0 \) . In particular, \( {\left\lbrack \mathcal{H}\left( {\bigotimes }^{p}{\mathbb{C}}^{n}\right) \right\rbrac... | Proof. We may assume \( p \leq q \) . Let \( 0 \neq T \in {\operatorname{Hom}}_{G}\left( {\mathcal{H}\left( {{\bigotimes }^{p}{\mathbb{C}}^{n}}\right) ,\mathcal{H}\left( {{\bigotimes }^{q}{\mathbb{C}}^{n}}\right) }\right) \) . Since \( - I \in G \) acts by \( {\left( -1\right) }^{p} \) on \( { \otimes }^{p}{\mathbb{C}}... | Yes |
Theorem 10.2.5. Let \( \mu \in \operatorname{Par}\left( {k, n}\right) \) . Then \( {\widetilde{W}}^{k}\left( \mu \right) \subset \mathcal{H}\left( {{\bigotimes }^{k}{\mathbb{C}}^{n}}\right) \) if and only if \( \mu \) is \( G \) -admissible. In this case \( {\widetilde{W}}^{k}\left( \mu \right) = {W}^{k}\left( \bar{\mu... | Proof. Write \( \mu = \mathop{\sum }\limits_{i}{m}_{i}{\varpi }_{i} \) and set\n\n\[ u = {\left( {w}_{1}\right) }^{\otimes {m}_{1}} \otimes \cdots \otimes {\left( {w}_{n}\right) }^{\otimes {m}_{n}}, \]\n\n(10.23)\n\nwhere \( {w}_{p} = {e}_{1} \land {e}_{2} \land \cdots \land {e}_{p} \) for \( 1 \leq p \leq n \) . Then ... | No |
Theorem 10.2.7. As a module for \( \operatorname{Sp}\left( {{\mathbb{C}}^{2l},\Omega }\right) \times {\mathfrak{S}}_{k} \), the space of \( \Omega \) -harmonic \( k \) - tensors has isotypic decomposition \[ \mathcal{H}\left( {{\bigotimes }^{k}{\mathbb{C}}^{2l},\Omega }\right) \cong {\bigoplus }_{\mu \in \operatorname{... | Proof. Suppose \( \lambda \) is a \( \mathfrak{b} \) -dominant weight and \( {W}^{k}\left( \lambda \right) \neq 0 \) . Then we must have \( \left| \lambda \right| = \) \( k - {2r} \) for some integer \( r \geq 0 \) . To see this, take \( 0 \neq v \in {W}^{k}\left( \lambda \right) \) and decompose \( v \) under \( \wide... | Yes |
Corollary 10.2.8. \( \left( {G = \mathbf{{Sp}}\left( {{\mathbb{C}}^{2l},\Omega }\right) }\right) \) Let \( \lambda \) be a dominant integral weight on \( \mathfrak{h} \) . Let \( k = \left| \lambda \right| \), so that \( \lambda \) determines a partition of \( k \) with at most \( l \) parts, and let \( A \) be a table... | Proof. By Proposition 9.3.5 we know that \( \mathbf{s}\left( A\right) \) projects \( {G}^{\lambda } \) onto a one-dimensional subspace spanned by a single \( \mathfrak{b} \) -extreme vector of weight \( \lambda \) and annihilates the spaces \( {G}^{\mu } \) for \( \mu \neq \lambda \) . Hence the corollary follows from ... | No |
Theorem 10.2.9. \( \left( {n = {2l} + 1}\right) \) As a module for \( \mathbf{O}\left( {{\mathbb{C}}^{n}, B}\right) \times {\mathfrak{S}}_{k} \), the space of \( B \) - harmonic k-tensors has isotypic decomposition\n\n\[ \mathcal{H}\left( {{\bigotimes }^{k}{\mathbb{C}}^{n}, B}\right) = {\bigoplus }_{\mu \in \mathbf{A}\... | Proof. Just as in the case of the symplectic group we will use Schur-Weyl duality and the characterization of \( G \) -admissible \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) -highest weights in Theorem 10.2.5. The new feature in this case is that for each \( \mathfrak{b} \) -dominant weight\n\n\[ \lambda = \mathop{... | Yes |
Corollary 10.2.11. \( \left( {G = \mathbf{O}\left( {{\mathbb{C}}^{n}, B}\right), n = {2l} + 1}\right) \) Let \( \lambda \) be a dominant integral weight on \( \mathfrak{h} \) . Let \( k = \left| \lambda \right| \) and \( m = \left| {\lambda }^{\natural }\right| \) . Let \( A \) (resp. \( {A}^{\natural } \) ) be a table... | Proof. The weights \( \lambda \) and \( {\lambda }^{\natural } \) are both in \( \mathbf{A}\left( {m, n}\right) \) . Hence we can use (10.25) and the same argument as in the case of the symplectic group (Corollary 10.2.8), noting that the integers \( k \) and \( m \) have opposite parity. | No |
Corollary 10.2.14. \( \left( {G = \mathbf{O}\left( {{\mathbb{C}}^{n}, B}\right), n = {2l}}\right) \) Let \( \lambda = \mathop{\sum }\limits_{{i = 1}}^{l}{\lambda }_{i}{\varepsilon }_{i} \) be a dominant integral weight on \( \mathfrak{h}\left( {{\lambda }_{1} \geq \cdots \geq {\lambda }_{l - 1} \geq {\lambda }_{l} \geq... | Proof. In Case (1) the weight \( \lambda \) is in \( {\mathbf{A}}_{ + }\left( {k, n}\right) \) and the weight \( {\lambda }^{\natural } \) is in \( {\mathbf{A}}_{ - }\left( {m, n}\right) \) . In Case (2) the weight \( \lambda \) is in \( {\mathbf{A}}_{0}\left( {k, n}\right) \) . Now use (10.34) and the same argument as... | Yes |
Lemma 10.3.2. \( {S}^{2}\left( {\mathop{\bigwedge }\limits^{2}{\mathbb{C}}^{n}}\right) \) is invariant under the Bianchi operator. | Proof. Let \( R \in {S}^{2}\left( {\mathop{\bigwedge }\limits^{2}{\mathbb{C}}^{n}}\right) \) . Since \( \left( {123}\right) = \left( {13}\right) \left( {12}\right) \) and \( \left( {321}\right) = \left( {23}\right) \left( {12}\right) \), we see that \( \sigma \left( {12}\right) {bR} = {b\sigma }\left( {12}\right) R = -... | Yes |
Proposition 10.3.3. If \( n \geq 2 \) then \( \operatorname{Curv}\left( {\mathbb{C}}^{n}\right) \cong {F}_{n}^{\left\lbrack 2,2\right\rbrack } \) as a \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) module. Hence\n\n\[ \n\dim \operatorname{Curv}\left( {\mathbb{C}}^{n}\right) = \frac{1}{12}{n}^{2}\left( {n + 1}\right) \... | Proof. Since \( {\beta }^{2} = \beta \) is a projection, we have the decomposition\n\n\[ \n{S}^{2}\left( {\mathop{\bigwedge }\limits^{2}{\mathbb{C}}^{n}}\right) = \operatorname{Ker}\left( \beta \right) \oplus \operatorname{Range}\left( \beta \right) ,\n\]\n\nwhich gives (10.52). Thus it remains to prove the first asser... | No |
Corollary 10.3.4. The space \( \operatorname{Curv}\left( {\mathbb{C}}^{n}\right) \) is the irreducible Weyl module for \( \mathbf{{GL}}\left( {n,\mathbb{C}}\right) \) corresponding to the tableau \( A = \frac{1}{2}\frac{3}{4} \) . | Proof. Since the representation \( {G}^{\left\lbrack 2,2\right\rbrack } \) of \( {\mathfrak{S}}_{4} \) has degree two, the normalized Young symmetrizer is \( {\mathbf{p}}_{A} = \left( {1/{12}}\right) \mathbf{c}\left( A\right) \mathbf{r}\left( A\right) \), where \( \mathbf{r}\left( A\right) = \left( {1 + \sigma \left( {... | Yes |
When \( n \geq 5 \) the space of curvature tensors on \( {\mathbb{C}}^{n} \) has the following structure relative to the orthogonal group: | 1. The space \( {\mathcal{H}}_{\mathrm{{sym}}}^{2}\left( {{\mathbb{C}}^{n}, Q}\right) \oslash Q \) corresponding to trace-zero Ricci curvature tensors is irreducible for \( \mathbf{{SO}}\left( {{\mathbb{C}}^{n}, Q}\right) \) and has highest weight \( 2{\varpi }_{1} \). 2. The space \( {\operatorname{Weyl}}_{Q}\left( {\... | Yes |
Lemma 10.4.1. Let \( S \in \operatorname{End}\left( {V \otimes V}\right) \) and set \( R = {\sigma S} \) . Then the set of operators \( \left\{ {{T}_{1}^{\left( n\right) }\left( S\right) ,\ldots ,{T}_{n - 1}^{\left( n\right) }\left( S\right) }\right\} \) satisfies the braid relations for all \( n = 3,4,\ldots \) if and... | To prove this result observe that the braid relations are satisfied by the operators \( {T}_{1}^{\left( n\right) }\left( S\right) ,\ldots ,{T}_{n - 1}^{\left( n\right) }\left( S\right) \) for all \( n \geq 3 \) if and only if the relations are satisfied when \( n = 3 \) . Then do the obvious calculation (which we leave... | No |
Lemma 10.4.2. Let \( R = {a\sigma } + {mbP} \) with \( a, b \in \mathbb{C} \) . Then\n\n\[ \left( {{R}_{12}{R}_{13}{R}_{23} - {R}_{23}{R}_{13}{R}_{12}}\right) {v}_{1} \otimes {v}_{2} \otimes {v}_{3} \]\n\n\[ = b\left( {{a}^{2} + {mab} + {b}^{2}}\right) \left\{ {\left( {{v}_{2},{v}_{3}}\right) \theta \otimes {v}_{1} - \... | This is proved by a rather lengthy but straightforward calculation that we leave as an exercise. | No |
Proposition 10.4.3. Let \( m = \dim V \geq 2 \) and take \( S \in {\operatorname{End}}_{G}\left( {V \otimes V}\right) \) . Set \( R = {\sigma S} \) . 1. Suppose either \( S = {\lambda I} \), or \( S = {\lambda \sigma } \) (for some \( \lambda \in {\mathbb{C}}^{ \times } \) ), or \( S = {aI} + {mbP} \) with \( a, b \in ... | Proof. For \( S \) as in (1), the operator \( R \) is of the form \( {\lambda I},{\lambda \sigma } \), or \( {a\sigma } + {mbP} \) with \( b \neq 0 \) and \( {a}^{2} + {mab} + {b}^{2} = 0 \) (note that \( {\sigma P} = P \) ). In all three cases \( R \) satisfies the Yang-Baxter equation (the first case is trivial, the ... | Yes |
Lemma 10.4.4. Suppose \( \left\{ {{g}_{1},\ldots ,{g}_{n}}\right\} \) generates a group \( G \) . Then there exists a unique surjective group homomorphism \( \psi : {F}_{n} \rightarrow G \) such that \( \psi \left( {x}_{i}\right) = {g}_{i} \) and \( \psi \left( {y}_{i}\right) = {g}_{i}^{-1} \) . | This follows directly from the definition of the multiplication in \( {F}_{n} \) . The group \( {F}_{n} \) is called the free group on \( n \) generators. | No |
1. If \( \beta \in {\mathcal{B}}_{n} \) then \( p\left( {b\left( m\right) ,{\left( \beta {\tau }_{n}\right) }_{n + 1}}\right) = p\left( {b\left( m\right) ,{\left( \beta {\tau }_{n}^{-1}\right) }_{n + 1}}\right) = p\left( {b\left( m\right) ,\beta }\right) \) . | The proof of (1) uses an idea of V. G. Turaev [144]. We first recall that if \( V \) is a finite-dimensional vector space then \( { \otimes }^{k}\operatorname{End}\left( V\right) \cong \operatorname{End}\left( {{ \otimes }^{k}V}\right) \), where\n\n\[ \left( {{A}_{1} \otimes {A}_{2} \otimes \cdots \otimes {A}_{k}}\righ... | Yes |
Theorem 11.1.1. Let \( G \) be a linear algebraic group. For every \( g \in G \) the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is a linear isomorphism from \( \operatorname{Lie}\left( G\right) \) onto \( T{\left( G\right) }_{g} \) . Hence \( G \) is a smooth algebraic set and \( \dim \operatorname{Lie}\left( G\r... | Proof. We first show that for fixed \( g \in G \), the map \( A \mapsto {\left( {X}_{A}\right) }_{g} \) is injective from \( \operatorname{Lie}\left( G\right) \) to \( T{\left( G\right) }_{g} \) . Suppose \( {\left( {X}_{A}\right) }_{g} = 0 \) . Then for \( x \in G \) and \( f \in \mathcal{O}\left\lbrack {\mathbf{{GL}}... | Yes |
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