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Proposition 31.12. Let \( \widetilde{t} \in \widetilde{T} \). Assume with the above identifications that the image of \( t \) in \( T \) is (31.2). Then the eigenvalues of \( \sigma \left( t\right) \) are the \( {2}^{n} \) values \( \prod {t}_{i}^{\pm 1/2} \) for an appropriate choice of the square roots. | Proof. Let \( t \in T \) be the element corresponding to \( \widetilde{t} \in \widetilde{T} \). By (31.12) we have, for \( r \in R \)\n\n\[ \sigma \left( \widetilde{t}\right) \omega \left( r\right) = \omega \left( {\rho \left( t\right) r}\right) \sigma \left( \widetilde{t}\right) .\n\]\n\n(31.13)\n\nHere \( \rho : \ope... | Yes |
Proposition 32.1. The representation \( {\operatorname{Ind}}_{B}^{G}\left( \chi \right) \) is of degree \( q + 1 \) . It is irreducible unless \( {\chi }_{1} = {\chi }_{2} \) . Moreover, it is isomorphic to \( {\operatorname{Ind}}_{B}^{G}\left( \mu \right) \) if and only if either \( {\chi }_{1} = {\mu }_{1} \) and \( ... | Proof. The index of \( B \) in \( G \) is easily seen to be \( q + 1 \), so this is the dimension of the induced representation. We recall that the vector space for the representation \( {\operatorname{Ind}}_{B}^{G}\left( \chi \right) \) consists of the space of functions \( f : G \rightarrow \mathbb{C} \) such that \(... | Yes |
Proposition 32.2. (Frobenius reciprocity, first version) Let \( H \) be a subgroup of \( G \) and let \( \left( {\pi, V}\right) \) be a representation of \( H \) . Let \( \left( {\sigma, U}\right) \) be a representation of \( G \) . Then\n\n\[ \n{\operatorname{Hom}}_{G}\left( {U,{V}^{G}}\right) \cong {\operatorname{Hom... | Proof. Given \( J \in {\operatorname{Hom}}_{G}\left( {U,{V}^{G}}\right) \), define \( j\left( u\right) = J\left( u\right) \left( 1\right) \) . We show that \( j \) is in \( {\operatorname{Hom}}_{H}\left( {{U}_{H}, V}\right) \) . Indeed, if \( h \in H \), we have\n\n\[ \nj\left( {\sigma \left( h\right) u}\right) = J\lef... | No |
Proposition 32.3. Let \( H \) be a subgroup of \( G \) and let \( \left( {\pi, V}\right) \) be a representation of \( H \) . If \( v \in V \), define \( \epsilon \left( v\right) : G \rightarrow V \) by\n\n\[ \epsilon \left( v\right) \left( g\right) = \left\{ \begin{matrix} \pi \left( g\right) v & \text{ if }g \in H, \\... | Proof. It is easy to check that \( \epsilon \left( v\right) \in {V}^{G} \) and that if \( h \in H \), then\n\n\[ \epsilon \left( {\pi \left( h\right) v}\right) = {\pi }^{G}\left( h\right) \epsilon \left( v\right) \]\n\nThus \( \epsilon \) is \( H \) -equivariant.\n\nWe prove that if \( f \in {V}^{G} \), then\n\n\[ f = ... | Yes |
Corollary 32.1. (Frobenius reciprocity, second version) If \( H \) is a subgroup of the finite group \( G \), and if \( \left( {\sigma, U}\right) \) and \( \left( {\pi, V}\right) \) are representations of \( G \) and \( H \), respectively, then \( {\operatorname{Hom}}_{G}\left( {{V}^{G}, U}\right) \cong {\operatorname{... | Proof. This is a direct restatement of Proposition 32.3. | Yes |
Theorem 32.1. (Mackey's theorem, geometric version) Suppose that \( G \) is a finite group, \( {H}_{1} \) and \( {H}_{2} \) subgroups, and \( \left( {{\pi }_{1},{V}_{1}}\right) \) and \( \left( {{\pi }_{2},{V}_{2}}\right) \) representations of \( {H}_{1} \) and \( {H}_{2} \), respectively. Then \( {\operatorname{Hom}}_... | Proof. Let \( \Delta \) satisfying (32.9) be given. It is easy to check, using (32.9) and the fact that \( f \in {V}_{1}^{G} \), that (32.10) is independent of the choice of coset representatives \( \gamma \) for \( G/{H}_{1} \) . Moreover, if \( {h}_{2} \in {H}_{2} \), then the variable change \( \gamma \rightarrow {h... | Yes |
Proposition 32.4. In the setting of Theorem 32.1, let \( \gamma \in G \) . Let \( {H}_{\gamma } = {H}_{2} \cap \) \( \gamma {H}_{1}{\gamma }^{-1} \) . Define two representations \( \left( {{\pi }_{1}^{\gamma },{V}_{1}}\right) \) and \( \left( {{\pi }_{2}^{\gamma },{V}_{2}}\right) \) of \( {H}_{\gamma } \) as follows. T... | Proof. If \( \Delta : G \rightarrow \operatorname{Hom}\left( {{V}_{1},{V}_{2}}\right) \) is associated with \( \Lambda \) as in Theorem 32.1, then \( \Delta \) is by assumption supported on \( {H}_{2}\gamma {H}_{1} \), and (32.9) implies that \( \Delta \) is determined by \( \delta = \Delta \left( \gamma \right) \) . T... | Yes |
Theorem 32.2. (Mackey's theorem, algebraic version) In the setting of Theorem 32.1, let \( {\gamma }_{1},\ldots ,{\gamma }_{h} \) be a complete set of representatives of the double cosets in \( {H}_{2} \smallsetminus G/{H}_{1} \) . With \( \gamma = {\gamma }_{i} \), let \( {\pi }_{i}^{\gamma } \) be as in Proposition 3... | Proof. If \( \Delta \) is as in Theorem 32.1, write \( \Delta = \mathop{\sum }\limits_{i}{\Delta }_{i} \), where\n\n\[ {\Delta }_{i}\left( g\right) = \left\{ \begin{matrix} \Delta \left( g\right) & \text{ if }g \in {H}_{2}{\gamma }_{i}{H}_{1} \\ 0 & \text{ otherwise. } \end{matrix}\right. \]\n\nThen \( {\Delta }_{i} \)... | Yes |
Assume that the ground field \( F \) is of characteristic zero. Let \( {\mathrm{H}}_{1} \) and \( {\mathrm{H}}_{2} \) be subgroups of \( G \) and let \( \left( {\pi, V}\right) \) be an irreducible representation of \( {H}_{1} \) . Let \( {\gamma }_{1},\ldots ,{\gamma }_{h} \) be a complete set of representatives of the... | \[ {\bigoplus }_{i = 1}^{h}{\operatorname{Ind}}_{{H}_{{\gamma }_{i}}}^{{H}_{2}}\left( {\pi }^{{\gamma }_{i}}\right) \] Proof. Since we are assuming that the characteristic of \( F \) is zero, representations are completely reducible and it is enough to show that the multiplicity of an irreducible representation \( \lef... | Yes |
Proposition 32.5. If \( R \) is a subring of \( A \) and \( V \) is a left \( R \) -module, let \( {V}^{\prime } \) be the left \( A \) -module \( A{ \otimes }_{R}V \) . We have a homomorphism \( i : V \rightarrow {V}^{\prime } \) of \( R \) -modules defined by \( i\left( v\right) = 1 \otimes v \) . If \( U \) is any l... | Proof. Suppose that \( J : {V}^{\prime } \rightarrow U \) is \( A \) -linear and satisfies \( j = J \circ i \) . Then\n\n\[ J\left( {a \otimes v}\right) = J\left( {a\left( {1 \otimes v}\right) }\right) = {aJ}\left( {1 \otimes v}\right) = {aJ}\left( {i\left( v\right) }\right) = {aj}\left( v\right) . \]\n\nSince \( {V}^{... | Yes |
Proposition 32.6. If \( R \) is a subring of \( A, U \) is a left \( A \) -module, and \( V \) is a left \( R \) -module, we have a natural isomorphism\n\n\[ \n{\operatorname{Hom}}_{R}\left( {V, U}\right) \cong {\operatorname{Hom}}_{A}\left( {A{ \otimes }_{R}V, U}\right) \n\] | Proof. This is a direct generalization of Proposition 11.1 (ii). It is also essentially equivalent to Proposition 32.5. Indeed, composition with \( i : V \rightarrow \) \( {V}^{\prime } = A{ \otimes }_{R}V \) is a map \( {\operatorname{Hom}}_{A}\left( {{V}^{\prime }, U}\right) \rightarrow {\operatorname{Hom}}_{R}\left(... | No |
Proposition 32.7. Suppose that \( H \) is a subgroup of \( G \) and \( V \) is an \( H \) -module. Then \( V \) is a module for the group ring \( F\left\lbrack H\right\rbrack \), which is a subring of \( F\left\lbrack G\right\rbrack \) . We have an isomorphism\n\n\[ \n{V}^{G} \cong F\left\lbrack G\right\rbrack { \otime... | Proof. Comparing Proposition 32.3 and Proposition 32.5, the \( G \) -modules \( {V}^{G} \) and \( F\left\lbrack G\right\rbrack { \otimes }_{F\left\lbrack H\right\rbrack }V \) satisfy the same universal property, so they are isomorphic. | Yes |
Proposition 32.8. Let \( \\left( {\\pi, V}\\right) \) be a complex representation of the subgroup \( H \) of the finite group \( G \) with character \( \\chi \) . Then the character of the induced representation \( {\\pi }^{G} \) is \( {\\chi }^{G} \) . | Proof. Let \( \\eta \) be the character of a representation \( \\left( {\\sigma, U}\\right) \) of \( G \) . We will prove that the class function \( {\\chi }^{G} \) satisfies Frobenius reciprocity in its classical form (32.4). This suffices because \( {\\chi }^{G} \) is determined by the inner product values \( \\left\... | Yes |
Proposition 33.1. The ring \( {\mathbb{Z}}_{\text{sym }}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) is generated as a \( \mathbb{Z} \) -algebra by \( {e}_{1},\ldots ,{e}_{n} \), and they are algebraically independent. Thus, \( {\mathbb{Z}}_{\text{sym }}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack = \)... | Proof. The fact that the \( {e}_{i} \) generate \( {\mathbb{Z}}_{\text{sym }}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) is Theorem 6.1 on p. 191 of Lang [116], and their algebraic independence is proved on p. 192 of that reference. The fact that \( {h}_{1},\ldots ,{h}_{n} \) also generate follows since (33.... | Yes |
Proposition 33.2. If \( V \) is an \( n \) -dimensional vector space and \( T : V \rightarrow V \) an endomorphism, and if \( {t}_{1},\ldots ,{t}_{n} \) are its eigenvalues with multiplicities (that is, each eigenvalue is listed with its multiplicity as a root of the characteristic polynomial), then\n\n\[ \operatorname... | Proof. First, assume that \( T \) is diagonalizable and that \( {v}_{1},\ldots ,{v}_{n} \) are its eigenvectors, so \( T{v}_{i} = {t}_{i}{v}_{i} \) . Then a basis of \( { \land }^{k}V \) consists of the vectors\n\n\[ {v}_{{i}_{1}} \land \cdots \land {v}_{{i}_{k}},\;1 \leq {i}_{1} < {i}_{2} < \cdots < {i}_{k} \leq n, \]... | Yes |
Theorem 33.1. Let \( f\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) be a symmetric polynomial with integer coefficients. Define a function \( {\psi }_{f} \) on \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) as follows. If \( {t}_{1},\ldots ,{t}_{n} \) are the eigenvalues of \( g \), let\n\n\[ \n{\psi }_{f}\left( g\right)... | Proof. Let us call a symmetric polynomial \( f \) constructible if \( {\psi }_{f} \) is a generalized character of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) . The generalized characters of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) form a ring since the direct sum and tensor product operations on \( \mathrm{... | Yes |
Theorem 33.2. Let \( G \) be a group, let \( \chi \) be a character of \( G \), and let \( k \) be a nonnegative integer. Then \( g \mapsto \chi \left( {g}^{k}\right) \) is a virtual character of \( G \) . | Proof. Let \( \chi \) be the character corresponding to the representation \( \pi : G \rightarrow \) \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) . If \( \psi \) is any generalized character of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), then \( \psi \circ \pi \) is a generalized character of \( G \) . We take ... | Yes |
Proposition 33.3. (Newton) The polynomials \( {p}_{k} \) generate \( {\mathbb{Q}}_{\text{sym }}\left\lbrack {{x}_{1},\ldots ,{x}_{n}}\right\rbrack \) as a \( \mathbb{Q} \) -algebra. | Proof. We will make use of the identity\n\n\[ \log \left( {1 + t}\right) = \mathop{\sum }\limits_{{k = 1}}^{\infty }\frac{{\left( -1\right) }^{k - 1}}{k}{t}^{k} \]\n\nReplacing \( t \) by \( t{x}_{i} \) in this identity, summing over the \( {x}_{i} \), and using (33.1), we see that\n\n\[ \log E\left( t\right) = \mathop... | Yes |
Proposition 34.1. Let \( V \) be a vector space over \( \mathbb{C} \). We have functorial isomorphisms\n\n\[ \land^k V \cong \left( \bigotimes^k V \right) \otimes_{\mathbb{C}\left\lbrack S_k \right\rbrack} \mathbb{C}_{\text{alt}}, \; \vee^k V \cong \left( \bigotimes^k V \right) \otimes_{\mathbb{C}\left\lbrack S_k \righ... | Proof. The proofs of these isomorphisms are similar. We will prove the first. It is sufficient to show that the right-hand side satisfies the universal property of the exterior \( k \) th power. We recall that this is the following property of \( \land^k V \). Given a vector space \( W \), a \( k \)-linear map \( f : V... | Yes |
Theorem 34.1. Let \( \rho : {S}_{k} \rightarrow \mathrm{{GL}}\left( {N}_{\rho }\right) \) be a representation. Let \( {V}_{\rho } \) be as in (34.2). There exists a homogeneous symmetric polynomial \( {s}_{\rho } \) of degree \( k \) in \( n \) variables such that if \( {\psi }_{\rho }\left( g\right) \) is the trace of... | Proof. First let us prove this for \( g \) restricted to the subgroup of diagonal matrices. Let \( {\xi }_{1},\ldots ,{\xi }_{n} \) be the standard basis of \( V \) . In other words, identifying \( V \) with \( {\mathbb{C}}^{n} \), let \( {\xi }_{i} = \left( {0,\ldots ,1,\ldots ,0}\right) \), where the 1 is in the \( i... | Yes |
Proposition 34.2. Let \( {\rho }_{i} : {S}_{k} \rightarrow \mathrm{{GL}}\left( {N}_{{\rho }_{i}}\right) \left( {i = 1,\ldots, h}\right) \) be the irreducible representations of \( {S}_{k} \) and let \( {d}_{1},\ldots ,{d}_{h} \) be their respective degrees. Then\n\n\[ \n{p}_{1}^{k} = \mathop{\sum }\limits_{i}{d}_{i}{s}... | Proof. If \( R \) is a ring and \( M \) a right \( R \) -module, then\n\n\[ \nM{ \otimes }_{R}R \cong M. \n\]\n\n(To prove this standard isomorphism, observe that \( m \otimes r \mapsto {mr} \) and \( m \mapsto m \otimes 1 \) are inverse maps between the two Abelian groups.) If \( M \) is an \( \left( {S, R}\right) \) ... | Yes |
Proposition 34.3. The homogeneous part \( {\Lambda }_{k}^{\left( n\right) } \) is a free Abelian group of rank equal to the number of partitions of \( k \) into no more than \( n \) parts. | Proof. Let \( {\lambda }^{\left( n\right) } \) be such a partition. Thus, \( {\lambda }^{\left( n\right) } = \left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \), where \( {\lambda }_{1} \geq \) \( {\lambda }_{2} \geq \cdots \geq {\lambda }_{n} \geq 0 \) and \( \mathop{\sum }\limits_{i}{\lambda }_{i} = k \) . Let\n... | Yes |
Proposition 34.4. The map \( {\operatorname{ch}}^{\left( n\right) } \) is a surjective homomorphism of graded rings. The map \( {\operatorname{ch}}_{k}^{\left( n\right) } \) in degree \( k \) is an isomorphism if \( n \geq k \) . | Proof. The main thing to check is that the group law \( \circ \) that was introduced in the ring \( \mathcal{R} \) corresponds to multiplication of polynomials. Indeed, let \( \theta \) and \( \rho \) be representations of \( {S}_{k} \) and \( {S}_{l} \), respectively. Then \( \theta \otimes \rho \) is an \( {S}_{k} \t... | Yes |
Proposition 34.5. \( \mathcal{R} \) is a polynomial ring in an infinite number of generators, \( \mathcal{R} = \mathbb{Z}\left\lbrack {{\mathbf{e}}_{1},{\mathbf{e}}_{2},{\mathbf{e}}_{3},\ldots }\right\rbrack = \mathbb{Z}\left\lbrack {{\mathbf{h}}_{1},{\mathbf{h}}_{2},{\mathbf{h}}_{3},\ldots }\right\rbrack \) . | Proof. To show that the \( {\mathbf{e}}_{i} \) generate \( \mathcal{R} \), it is sufficient to show that the ring they generate contains an arbitrary element \( u \) of \( {\mathcal{R}}_{k} \) for any fixed \( k \) . Take \( n \geq k \) . Since \( {e}_{1},\ldots ,{e}_{n} \) generate the ring \( {\Lambda }^{\left( n\rig... | Yes |
Proposition 34.6. We have \( {r}_{n} \circ {\mathrm{{ch}}}^{\left( n + 1\right) } = {\mathrm{{ch}}}^{\left( n\right) } \) as maps \( \mathcal{R} \rightarrow {\Lambda }^{\left( n\right) } \) . | Proof. It is enough to check this on \( {\mathbf{e}}_{1},{\mathbf{e}}_{2},\ldots \) since they generate \( \mathcal{R} \) by Proposition 33.1. Both maps send \( {\mathbf{e}}_{k} \rightarrow {e}_{k} \) if \( k \leq n \), and \( {\mathbf{e}}_{k} \rightarrow 0 \) if \( k > n \) . | Yes |
Theorem 34.3. The rings \( \mathcal{R} \) and \( \Lambda \) admit automorphisms of order 2 that interchange \( {\mathbf{e}}_{i} \leftrightarrow {\mathbf{h}}_{i} \) and \( {e}_{i} \leftrightarrow {h}_{i} \) . | Proof. Of course, it does not matter which ring we work in. Since \( \Lambda \cong \) \( \mathbb{Z}\left\lbrack {{e}_{1},{e}_{2},{e}_{3},\ldots }\right\rbrack \), and since the \( {e}_{i} \) are algebraically independent, if \( {u}_{1},{u}_{2},\ldots \) are arbitrarily elements of \( \Lambda \), there exists a unique r... | Yes |
Proposition 35.1. Let \( A \) be a matrix of determinant 1, and let \( B = {}^{t}{A}^{-1} \) . Each minor of \( A \) equals \( \pm \) the complementary minor of \( B \) . | Proof. Let us show how to prove this fact using exterior algebra. Suppose that \( A \) is an \( N \times N \) matrix. Let \( V = {\mathbb{C}}^{N} \) . Then \( { \land }^{N}V \) is one-dimensional, and we fix an isomorphism \( \eta : { \land }^{N}V \rightarrow \mathbb{C} \) . If \( 1 \leq k \leq N \), and if \( A : V \r... | Yes |
Proposition 35.2. Suppose that \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{r}}\right) \) and \( \mu = \left( {{\mu }_{1},\ldots ,{\mu }_{s}}\right) \) are conjugate partitions of \( k \) . Then the \( r + s \) numbers\n\n\[ s + i - {\lambda }_{i},\;\left( {i = 1,\ldots, r}\right) ,\]\n\n\[ s - j + {\mu }_{j... | Proof. First note that the \( r + s \) integers all lie between 0 and \( r + s \) . Indeed, if \( 1 \leq i \leq r \), then\n\n\[ 0 \leq s + i - {\lambda }_{i} \leq s + r \]\n\nbecause \( s \) is greater than or equal to the length \( l\left( \mu \right) = {\lambda }_{1} \geq {\lambda }_{i} \), so \( s + i - {\lambda }_... | Yes |
Proposition 35.3. We have\n\n\[ \n{\mathbf{h}}_{k} - {\mathbf{e}}_{1}{\mathbf{h}}_{k - 1} + {\mathbf{e}}_{2}{\mathbf{h}}_{k - 2} - \cdots + {\left( -1\right) }^{k}{\mathbf{e}}_{k} = 0 \n\]\n\n(35.2)\n\nif \( k \geq 1 \) . | Proof. Choose \( n \geq k \) so that the characteristic map \( {\operatorname{ch}}^{\left( n\right) } : {\mathcal{R}}_{k} \rightarrow {\Lambda }_{k}^{\left( n\right) } \) is injective. It is then sufficient to prove that \( {\operatorname{ch}}^{\left( n\right) } \) annihilates the left-hand side. Since \( {\operatornam... | Yes |
Proposition 35.4. Let \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{r}}\right) \) and \( \mu = \left( {{\mu }_{1},\ldots ,{\mu }_{s}}\right) \) be conjugate partitions of \( k \) . Then\n\n\[ \det {\left( {h}_{{\lambda }_{i} - i + j}\right) }_{1 \leq i, j \leq r} = \pm \det \left( {e}_{{\mu }_{i} - i + j}\rig... | Proof. We may interpret (33.3) as saying that the Toeplitz matrix\n\n\[ \left( \begin{matrix} {h}_{0} & {h}_{1} & \cdots & {h}_{r + s - 1} \\ & {h}_{0} & \cdots & {h}_{r + s - 2} \\ & & \ddots & \vdots \\ & & & {h}_{0} \end{matrix}\right) \]\n\n(35.4)\n\nis the transpose inverse of\n\n\[ \left( \begin{matrix} {e}_{0} &... | Yes |
Corollary 35.1. If \( \lambda \) and \( \mu \) are partitions of \( k \), then we have \( {\mu }^{t} \succcurlyeq {\lambda }^{t} \) if and only if \( \lambda \succcurlyeq \mu \) . | Proof. This is equivalent to the statement that \( {\mu }^{t} \succcurlyeq \lambda \) if and only if \( {\lambda }^{t} \succcurlyeq \mu \) . In this form, this is contained in the preceding proposition from the identity (35.6) together with the characterization of the nonvanishing of that inner product. Of course, one ... | No |
Theorem 35.2. If \( \lambda \) and \( \mu \) are conjugate partitions, and if \( \iota \) is the involution of Theorem 34.3, then \( {}^{\iota }{\mathbf{s}}_{\lambda } = {\mathbf{s}}_{\mu } \) and \( {}^{\iota }{s}_{\lambda } = {s}_{\mu } \). | Proof. Since \( {}^{\iota }{\mathbf{h}}_{\lambda } = {\mathbf{e}}_{\lambda } \) and \( {}^{\iota }{\mathbf{e}}_{\lambda } = {\mathbf{h}}_{\lambda } \), this follows from the Jacobi-Trudi identity. | Yes |
Theorem 36.1. Assume that \( n \geq l\left( \lambda \right) \) . We have\n\n\[ \n{s}_{\lambda }\left( {{x}_{1},\ldots ,{x}_{n}}\right) = \frac{\left| \begin{matrix} {x}_{1}^{{\lambda }_{1} + n - 1} & {x}_{2}^{{\lambda }_{1} + n - 1} & \cdots & {x}_{n}^{{\lambda }_{1} + n - 1} \\ {x}_{1}^{{\lambda }_{2} + n - 2} & {x}_{... | Proof. Let \( {e}_{k}^{\left( i\right) } \) be the \( k \) th elementary symmetric matrix in \( n - 1 \) variables\n\n\[ \n{x}_{1},\ldots ,{x}_{i - i},{x}_{i + 1},\ldots ,{x}_{n} \]\n\nomitting \( {x}_{i} \) . We have, using (33.1) and (33.2) and omitting one variable in (33.1),\n\n\[ \n\mathop{\sum }\limits_{{k = 0}}^... | Yes |
Proposition 36.1. Let \( U \) and \( W \) be vector spaces over a field of characteristic zero and let \( B : U \times \cdots \times U \rightarrow W \) be a symmetric \( k \) -linear map. Let \( Q \) : \( U \rightarrow W \) be the function \( Q\left( u\right) = B\left( {u,\ldots, u}\right) \) . If \( {u}_{1},\ldots ,{u... | Proof. Expanding \( Q\left( {u}_{S}\right) = B\left( {{u}_{S},\ldots ,{u}_{S}}\right) \) and using the \( k \) -linearity of \( B \), we have \[ Q\left( {u}_{S}\right) = \mathop{\sum }\limits_{{{i}_{1},\ldots ,{i}_{k} \in S}}B\left( {{u}_{{i}_{1}},{u}_{{i}_{2}},\ldots ,{u}_{{i}_{k}}}\right) . \] Therefore, \[ \mathop{\... | Yes |
Proposition 36.2. Suppose that \( n \geq l\left( \lambda \right) \). Let\n\n\[{\lambda }^{\prime } = \left( {{\lambda }_{1} - {\lambda }_{n},{\lambda }_{2} - {\lambda }_{n},\ldots ,{\lambda }_{n - 1} - {\lambda }_{n},0}\right) .\n\]\n\nIn the ring \( {\Lambda }^{\left( n\right) } \) of symmetric polynomials in \( n \) ... | Proof. It follows from (36.1) that \( {s}_{\lambda }\left( {{x}_{1},\ldots ,{x}_{n}}\right) \) is divisible by \( {\left( {x}_{1}\cdots {x}_{n}\right) }^{{\lambda }_{n}} \). Indeed, each entry of the first column of the matrix in the numerator is divisible by \( {x}_{1}^{{\lambda }_{n}} \), so we may pull \( {x}_{1}^{{... | Yes |
Proposition 36.3. Let \( \pi \) be a finite-dimensional irreducible representation of \( \mathrm{U}\left( n\right) \) . Then \( \pi \) is isomorphic to the restriction of \( {\pi }_{\lambda }^{\mathrm{{GL}}\left( n\right) } \) for some \( \lambda \) . | Proof. Let \( G = \mathrm{U}\left( n\right) \) . By Schur orthogonality, it is enough to show that the characters of the \( {\pi }_{\lambda } = {\pi }_{\lambda }^{\mathrm{{GL}}\left( n\right) } \) are dense in the space of class functions in \( {L}^{2}\left( G\right) \) . We refer to a symmetric polynomial in \( {\alph... | Yes |
Lemma 36.1. If \( f \) is an analytic function on \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), then \( f \) is determined by its restriction to \( \mathrm{U}\left( n\right) \) . | Proof. We show that if \( {\left. f\right| }_{\mathrm{U}\left( n\right) } = 0 \) then \( f = 0 \) . Let \( \mathfrak{g} \) be the Lie algebra of \( \mathrm{U}\left( n\right) \) of consisting of skew-Hermitian matrices. Then the exponential map \( \exp : \mathfrak{g} \rightarrow \mathrm{U}\left( n\right) \) is surjectiv... | Yes |
Proposition 36.4. Let \( {\pi }_{1} \) and \( {\pi }_{2} \) be analytic representations of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \). If \( {\pi }_{1} \) and \( {\pi }_{2} \) have isomorphic restrictions to \( \mathrm{U}\left( n\right) \), they are isomorphic. | Proof. We may assume that \( {\pi }_{1} \) and \( {\pi }_{2} \) act on the same complex vector space \( V \), and that \( {\pi }_{1}\left( g\right) = {\pi }_{2}\left( g\right) \) when \( g \in \mathrm{U}\left( n\right) \). Applying Lemma 36.1 to the matrix coefficients of \( {\pi }_{1} \) and \( {\pi }_{2} \) it follow... | Yes |
Theorem 36.3. Every finite-dimensional representation of the group \( \mathrm{U}\left( n\right) \) extends uniquely to an analytic representation of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) . The irreducible complex representations of \( \mathrm{U}\left( n\right) \), or equivalently the irreducible analytic comp... | Proof. The fact that irreducible representations of \( \mathrm{U}\left( n\right) \) extend to analytic representations follows from the fact that such a representation is a \( {\pi }_{\lambda }^{\mathrm{{GL}}\left( n\right) } \) , proved in Proposition 36.3. Since \( \mathrm{U}\left( n\right) \) is compact, each repres... | Yes |
Proposition 36.5. Suppose that \( \lambda \) is a partition and \( l\left( \lambda \right) > n \) . Then we have \( {s}_{\lambda }\left( {{x}_{1},\ldots ,{x}_{n}}\right) = 0 \) in the ring \( {\Lambda }^{\left( n\right) } \) . | Proof. If \( N = l\left( \lambda \right) \), then \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{N}}\right) \), where \( {\lambda }_{N} > 0 \) and \( N > n \) . Apply the homomorphism \( {r}_{N - 1} \) defined by (34.9), noting that \( {r}_{N - 1}\left( {e}_{N}\right) = 0 \), since \( {e}_{N} \) is divisible b... | Yes |
Theorem 36.4. If \( \lambda \) is a partition of \( k \) let \( {\rho }_{\lambda } \) denote the irreducible representation of \( {S}_{k} \) affording the character \( {\mathbf{s}}_{\lambda } \) constructed in Theorem 35.1. If, moreover, \( l\left( \lambda \right) \leq n \), let \( {\pi }_{\lambda } \) denote the irred... | Proof. Most of this was proved in the proof of Theorem 36.2. Particularly, we saw there that each irreducible representation of \( {S}_{k} \) occurring in (36.5) occurs at most once and is paired with an irreducible representation of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) . If \( l\left( \lambda \right) \leq n... | Yes |
Proposition 37.1. Let \( {m}_{r} \) be the number of \( i \) such that \( {\lambda }_{i} = r \) . Then\n\n\[ \n{z}_{\lambda } = \mathop{\prod }\limits_{{r = 1}}^{k}{r}^{{m}_{r}}{m}_{r}! \n\] | Proof. \( {z}_{\lambda } \) is the order of the centralizer of a representative element \( g \in {\mathcal{C}}_{\lambda } \) . This centralizer is easily described.\n\nFirst, we consider the case where \( g \) contains only cycles of length \( r \) in its decomposition into disjoint cycles. In this case (denoting \( {m... | Yes |
Proposition 37.2. The character values of the irreducible representations of \( {S}_{k} \) are rational integers. | Proof. Using the Jacobi-Trudi identity (Theorem 35.1), \( {\mathbf{s}}_{\lambda } \) is a sum of terms of the form \( \pm {\mathbf{h}}_{\mu } \) for various partitions \( \mu \) . Each \( {\mathbf{h}}_{\mu } \) is the character induced from the trivial character of \( {S}_{\mu } \), so it has integer values. | Yes |
Proposition 37.3. If \( g \in {\mathcal{C}}_{\lambda } \), then \( \left\langle {{\mathbf{s}}_{\mu },{\mathbf{p}}_{\lambda }}\right\rangle = {\mathbf{s}}_{\mu }\left( g\right) \) . | Proof. We have\n\n\[ \left\langle {{\mathbf{s}}_{\mu },{\mathbf{p}}_{\lambda }}\right\rangle = \frac{1}{\left| {S}_{k}\right| }\mathop{\sum }\limits_{{x \in {\mathcal{C}}_{\lambda }}}{z}_{\lambda }{\mathbf{s}}_{\mu }\left( x\right) \]\n\nThe summand is constant on \( {\mathcal{C}}_{\lambda } \) and equals \( {z}_{\lamb... | Yes |
Proposition 37.4. If \( \lambda \) is a partition of \( k \), then \( {\mathbf{p}}_{\lambda } \in {\mathcal{R}}_{k} \) . | Proof. The inner products \( \left\langle {{\mathbf{p}}_{\lambda },{\mathbf{s}}_{\mu }}\right\rangle \) are rational integers by Propositions 37.2 and 37.3. By Schur orthogonality, we have \( {\mathbf{p}}_{\lambda } = \mathop{\sum }\limits_{\mu }\left\langle {{\mathbf{p}}_{\lambda },{\mathbf{s}}_{\mu }}\right\rangle {\... | Yes |
Proposition 37.5. If \( h = l\left( \lambda \right) \), so \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{h}}\right) \) and \( {\lambda }_{h} > 0 \), then\n\n\[{\mathbf{p}}_{\lambda } = {\mathbf{p}}_{{\lambda }_{1}}{\mathbf{p}}_{{\lambda }_{2}}\cdots {\mathbf{p}}_{{\lambda }_{h}}\] | Proof. From the definitions, \( {\mathbf{p}}_{{\lambda }_{1}}\cdots {\mathbf{p}}_{{\lambda }_{h}} \) is induced from the class function \( f \) on the subgroup \( {S}_{\lambda } \) of \( {S}_{k} \) which has a value on \( \left( {{\sigma }_{1},\ldots ,{\sigma }_{h}}\right) \) that is\n\n\[ \left\{ \begin{matrix} {\lamb... | Yes |
Proposition 37.6. We have\n\n\[ k{\mathbf{h}}_{k} = \mathop{\sum }\limits_{{r = 1}}^{k}{\mathbf{p}}_{r}{\mathbf{h}}_{k - r} \]\n\n(37.3) | Proof. Let \( \lambda \) be a partition of \( k \) . Let \( {m}_{s} \) be the number of \( {\lambda }_{i} \) equal to \( s \) . We will prove\n\n\[ \left\langle {{\mathbf{p}}_{r}{\mathbf{h}}_{k - r},{\mathbf{p}}_{\lambda }}\right\rangle = r{m}_{r} \]\n\n(37.4)\n\nBy Frobenius reciprocity, this inner product is \( {\lef... | Yes |
Proposition 37.7. We have\n\n\[ k{h}_{k} = \mathop{\sum }\limits_{{r = 1}}^{k}{p}_{r}{h}_{k - r} \] | Proof. We recall from (33.2) that\n\n\[ \mathop{\sum }\limits_{{k = 0}}^{\infty }{h}_{k}{t}^{k} = \mathop{\prod }\limits_{{i = 1}}^{n}{\left( 1 - {x}_{i}t\right) }^{-1} \]\n\nwhich we differentiate logarithmically to obtain\n\n\[ \frac{\mathop{\sum }\limits_{{k = 0}}^{\infty }k{h}_{k}{t}^{k - 1}}{\mathop{\sum }\limits_... | Yes |
Theorem 37.2. Express each symmetric polynomial \( {p}_{\lambda } \) as a linear combination of the \( {s}_{\mu } \) :\n\n\[ \n{p}_{\lambda } = \mathop{\sum }\limits_{\mu }{c}_{\lambda \mu }{s}_{\mu } \n\]\n\nThen the coefficient \( {c}_{\lambda \mu } \) is the value of the irreducible character \( {\mathbf{s}}_{\mu } ... | Proof. Since \( n \geq k \), ch : \( {\mathcal{R}}_{k} \rightarrow {\Lambda }_{k} \) is injective, and it follows that\n\n\[ \n{\mathbf{p}}_{\lambda } = \mathop{\sum }\limits_{\mu }{c}_{\lambda \mu }{\mathbf{s}}_{\mu } \n\]\n\nTaking the inner product of this relation with \( {\mathbf{s}}_{\mu } \), we see that\n\n\[ \... | Yes |
Theorem 37.3. (Frobenius) Let \( \mu \) be a partition of \( k \) of length \( \leq n \), and let \( \lambda \) be another partition of \( k \) . Let \( {c}_{\lambda \mu } \) be the value of the character \( {\mathbf{s}}_{\mu } \) on elements of the conjugacy class \( {\mathcal{C}}_{\lambda } \) . Then \( {c}_{\lambda ... | Proof. By Theorem 37.2, we have \( {p}_{\lambda } = \mathop{\sum }\limits_{\mu }{c}_{\lambda \mu }{s}_{\mu } \), and by (36.1) this means that\n\n\[ \n{p}_{\lambda }\Delta = \mathop{\sum }\limits_{\mu }{c}_{\lambda \mu }\det \left( {x}_{j}^{{\mu }_{i} + n - i}\right) , \n\]\n\nthe determinant being the determinant in t... | Yes |
Lemma 37.1. Let \( H \) be a subgroup of the finite group \( G \). Let \( \chi \) be a character of \( H \), and let \( \rho \) be a one-dimensional character of \( G \), which we may restrict to \( H \). The induced character \( {\left( \rho \chi \right) }^{G} \) equals \( \rho {\chi }^{G} \). | Proof. This may be proved either directly from the definition of the induced representation or by using (32.15). | No |
Theorem 37.4. If \( \mathbf{f} \) is a class function on \( {S}_{k} \), its involute ’ \( \mathbf{f} \) is the result of multiplying \( \mathbf{f} \) by the alternating character \( \varepsilon \) of \( {S}_{k} \) . | Proof. Let us denote by \( \tau : {\mathcal{R}}_{k} \rightarrow {\mathcal{R}}_{k} \) the linear map that takes a class function \( \mathbf{f} \) on \( {S}_{k} \) and multiplies it by \( \varepsilon \), and assemble the \( \tau \) in different degrees to a linear map of \( \mathcal{R} \) to itself. We want to prove that... | Yes |
Proposition 38.1. Let \( k \) be a nonnegative integer. Then we have the following identity in the ring \( {\Lambda }^{\left( N\right) } \) of symmetric polynomials.\n\n\[ \mathop{\sum }\limits_{{\lambda \text{ a partition of }k}}{z}_{\lambda }^{-1}{p}_{\lambda } = {h}_{k} \] | Proof. In view of Theorem 37.1 it is sufficient to show in \( \mathcal{R} \) that\n\n\[ \mathop{\sum }\limits_{{\lambda \text{ a partition of }k}}{z}_{\lambda }^{-1}{\mathbf{p}}_{\lambda } = {\mathbf{h}}_{k} \]\n\nWe consider both sides as functions on \( {S}_{k} \) . By definition, \( {\mathbf{p}}_{\lambda } \) is the... | Yes |
Proposition 38.2. Let \( k \) be a nonnegative integer. Then\n\n\[ \mathop{\sum }\limits_{{\lambda \text{ a partition of }k}}{s}_{\lambda }\left( \alpha \right) {s}_{\lambda }\left( \beta \right) = \mathop{\sum }\limits_{{\lambda \text{ a partition of }k}}{z}_{\lambda }^{-1}{p}_{\lambda }\left( \alpha \right) {p}_{\lam... | Proof. Both sides polynomials in the \( {\alpha }_{i} \) and \( {\beta }_{i} \) that are symmetric and homogeneous of degree \( k \) in either set of variables. Use the Frobenius-Schur duality to transfer the function on the right-hand side to a function on \( {S}_{k} \times {S}_{k} \) . In view of Theorem 37.1 this is... | Yes |
Theorem 38.1. (Cauchy) Suppose \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) and \( {\beta }_{1},\ldots ,{\beta }_{m} \) are complex numbers of absolute value \( < 1 \) . Then\n\n\[ \mathop{\prod }\limits_{{i = 1}}^{n}\mathop{\prod }\limits_{{j = 1}}^{m}{\left( 1 - {\alpha }_{i}{\beta }_{j}\right) }^{-1} = \mathop{\sum }\l... | Proof. Using (33.2) in the \( {nm} \) variables \( {\alpha }_{i}{\beta }_{j} \), the left-hand side equals\n\n\[ \mathop{\sum }\limits_{{k = 0}}{h}_{k}\left( {{\alpha }_{i}{\beta }_{j}}\right) \]\n\nso it is sufficient to show\n\n\[ \mathop{\sum }\limits_{{\lambda \text{ a partition of }k}}{s}_{\lambda }\left( {{\alpha... | Yes |
Proposition 38.3. Let \( G = {\mathrm{{GL}}}_{n}\left( \mathbb{C}\right) \times {\mathrm{{GL}}}_{m}\left( \mathbb{C}\right) \) acting on the tensor product \( \Omega = {\mathbb{C}}^{n} \otimes {\mathbb{C}}^{m} \) of the standard modules of \( {\mathrm{{GL}}}_{n}\left( \mathbb{C}\right) \) and \( {\mathrm{{GL}}}_{m}\lef... | Proof. If \( g \) has eigenvalues \( {\alpha }_{i} \) and \( h \) has eigenvalues \( {\beta }_{j} \), then \( \left( {g, h}\right) \) has eigenvalues \( {\alpha }_{i}{\beta }_{j} \) on \( \Omega \), hence has trace \( {h}_{k}\left( {{\alpha }_{i}{\beta }_{j}}\right) \) on \( { \vee }^{k}\Omega \) . By the Cauchy identi... | Yes |
Theorem 38.2. Suppose \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) and \( {\beta }_{1},\ldots ,{\beta }_{m} \) are complex numbers of absolute value \( < 1 \) . Then\n\n\[ \mathop{\prod }\limits_{{i = 1}}^{n}\mathop{\prod }\limits_{{j = 1}}^{m}\left( {1 + {\alpha }_{i}{\beta }_{j}}\right) = \mathop{\sum }\limits_{\lambda ... | Proof. Let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be fixed complex numbers, and let \( {\Lambda }^{\left( m\right) } \) be the ring of symmetric polynomials in \( {\beta }_{1},\ldots ,{\beta }_{m} \) with integer coefficients. We recall from Theorems 34.3 and 35.2 that \( \Lambda \) has an involution \( \iota \) tha... | Yes |
Proposition 38.4. Let there be given a see-saw (38.9). Let \( {\pi }_{i}^{G} \) and \( {\pi }_{i}^{{H}^{\prime }} \) be corresponding representations of \( G \) and \( {H}^{\prime } \), and let \( {\pi }_{j}^{{G}^{\prime }} \) and \( {\pi }_{j}^{H} \) be corresponding representations of \( {G}^{\prime } \) and \( H \) ... | Proof. We may express the correspondences as follows:\n\n\[ \n{\left. \Theta \right| }_{G \times {H}^{\prime }} = {\bigoplus }_{i \in I}{\pi }_{i}^{G} \otimes {\pi }_{i}^{{H}^{\prime }},{\left. \;\Theta \right| }_{H \times {G}^{\prime }} = {\bigoplus }_{j \in J}{\pi }_{j}^{H} \otimes {\pi }_{j}^{{G}^{\prime }}, \n\]\n\... | Yes |
Proposition 39.1. Let \( \mathbf{f} \) be a class function on \( {S}_{k} \) . Write \( \mathbf{f} = \mathop{\sum }\limits_{\lambda }{c}_{\lambda }{\mathbf{s}}_{\lambda } \), where the sum is over the partitions of \( k \) . Then\n\n\[ \n{\left| \mathbf{f}\right| }^{2} = \mathop{\sum }\limits_{\lambda }{\left| {c}_{\lam... | Proof. The \( {\mathbf{s}}_{\lambda } \) are orthonormal by Schur orthogonality, so \( {\left| \mathbf{f}\right| }^{2} = \sum {\left| {c}_{\lambda }\right| }^{2} \) . By Theorem 36.2, \( {\mathrm{{Ch}}}^{\left( n\right) }\left( {\mathbf{s}}_{\lambda }\right) \) are distinct irreducible characters when \( \lambda \) run... | Yes |
Theorem 39.1. The map \( {\mathrm{{Ch}}}^{\left( n\right) } \) is a contraction if \( n < k \) and an isometry if \( n \geq k \) . In other words, if \( \mathbf{f} \) is a class function on \( {S}_{k} \) , | \[ \left| {{\operatorname{Ch}}^{\left( n\right) }\left( \mathbf{f}\right) }\right| \leq \left| \mathbf{f}\right| \] with equality when \( n \geq k \) . Proof. This follows immediately from Proposition 39.1 since if \( n \geq k \) every partition of \( k \) has length \( \leq n \) . | Yes |
Proposition 39.2. Let \( k, l \geq 0 \) . Then\n\n\[ \n{\int }_{\mathrm{U}\left( n\right) }\operatorname{tr}{\left( g\right) }^{k}\overline{\operatorname{tr}{\left( g\right) }^{l}}{dg} = 0\;\text{ if }k \neq l, \]\n\nwhile\n\n\[ \n{\int }_{\mathrm{U}\left( n\right) }{\left| \operatorname{tr}\left( g\right) \right| }^{2... | Proof. If \( k \neq l \), then the variable change \( g \rightarrow {e}^{i\theta }g \) multiplies the left-hand side by \( {e}^{i\left( {k - l}\right) \theta } \neq 1 \) for \( \theta \) in general position, so the integral vanishes.\n\nAssume that \( k = l \) . We show that\n\n\[ \n{\int }_{\mathrm{U}\left( n\right) }... | Yes |
Theorem 39.2. Suppose that \( \\phi \\left( z\\right) \) is a polynomial in \( z \) and \( \\bar{z} \) of degree \( \\leq {2n} \) . Then\n\n\[ \n{\\int }_{\\mathrm{U}\\left( n\\right) }\\phi \\left( {\\operatorname{tr}\\left( g\\right) }\\right) {dg} = {\\int }_{\\mathbb{C}}\\phi \\left( z\\right) {d\\mu }\\left( z\\ri... | Proof. It is sufficient to prove this if \( \\phi \\left( z\\right) = {z}^{k}{\\bar{z}}^{l} \) . If \( \\deg \\left( \\phi \\right) \\leq {2n} \), then \( k + l \\leq \) \( {2n} \) so either \( k \\neq l \) or both \( k, l \\leq n \), and in either case Proposition 39.2 implies that the left-hand side equals 0 if \( k ... | Yes |
Theorem 39.3. Let \( {\nu }_{1},{\nu }_{2},{\nu }_{3},\ldots \) and \( \nu \) be probability measures on \( {\mathbb{R}}^{N} \) . Suppose that the characteristic functions \( \widehat{{\nu }_{i}}\left( {{y}_{1},\ldots ,{y}_{N}}\right) \rightarrow \widehat{\nu }\left( {{y}_{1},\ldots ,{y}_{N}}\right) \) pointwise for al... | Proof omitted. A proof may be found in Billingsley [18], Theorem 26.3 (when \( N = 1 \) ) and Sect. 28 (for general \( N \) ). The precise statement we need is on p. 383 before Theorem 29.4. | No |
Proposition 39.3. The functions \( {\widehat{\mu }}_{n} \) converge uniformly on compact subsets of \( \mathbb{C} \) to \( \widehat{\mu } \) . | Proof. The function \( \widehat{\mu } \) is easily computed. As the Fourier transform of a Gaussian distribution, \( \widehat{\mu } \) is also Gaussian and in fact \( \widehat{\mu }\left( w\right) = {e}^{-{\left| w\right| }^{2}} \) . We write this as a power series:\n\n\[ \widehat{\mu }\left( w\right) = F\left( \left| ... | Yes |
Corollary 39.1. The measures \( {\mu }_{n} \) converge weakly to \( \mu \) . | Proof. This follows immediately from the criterion of Theorem 39.3. | No |
Theorem 39.4. (Diaconis and Shahshahani) The joint probability distribution of the \( \left( {\operatorname{tr}\left( g\right) ,\operatorname{tr}\left( {g}^{2}\right) ,\ldots ,\operatorname{tr}\left( {g}^{r}\right) }\right) \) near \( \left( {{z}_{1},\ldots ,{z}_{r}}\right) \in {\mathbb{C}}^{r} \) is a measure weakly c... | Proof. Indeed, this follows along the lines of Corollary 39.1 using the fact that the moments of the measure (39.8)\n\n\[ {\int }_{\mathbb{C}}{\left| {z}_{1}\right| }^{2{k}_{1}}{\left| {z}_{2}\right| }^{2{k}_{2}}\cdots {\left| {z}_{r}\right| }^{2{k}_{r}}\mathop{\prod }\limits_{{j = 1}}^{r}\frac{1}{j}\pi {e}^{-\pi {\lef... | Yes |
Theorem 39.5. (Keating and Snaith) Let \( k \) be a nonnegative integer. Then\n\n\[ \n{\int }_{\mathrm{U}\left( n\right) }{\left| \det \left( g - I\right) \right| }^{2k}{dg} = \mathop{\prod }\limits_{{j = 0}}^{{n - 1}}\frac{j!\left( {j + {2k}}\right) !}{\left( {j + k}\right) {!}^{2}} \n\] | Proof. Let \( {t}_{1},\ldots ,{t}_{k} \) and \( {u}_{1},\ldots ,{u}_{k} \) be complex numbers. We will show that\n\n\[ \n{\int }_{\mathrm{U}\left( n\right) }\mathop{\prod }\limits_{{i = 1}}^{k}\left( {1 + {t}_{i}\det \left( g\right) }\right) \left( {{u}_{i} + \det {\left( g\right) }^{-1}}\right) {dg} = {s}_{\left( {n}^... | Yes |
Proposition 40.1. Let \( \\lambda \) be a partition of \( k \), and let \( \\mu \) be a partition of \( k - 1 \) . Then\n\n\[ \n\\left\\langle {{\\mathbf{s}}_{\\lambda },{\\mathbf{s}}_{\\mu }{\\mathbf{e}}_{1}}\\right\\rangle = \\left\\{ \\begin{array}{l} 1\\text{ if }\\lambda \\supset \\mu \\\\ 0\\text{ otherwise. } \\... | Proof. Applying ch, it is sufficient to show that\n\n\[ \n{e}_{1}{s}_{\\mu } = \\mathop{\\sum }\\limits_{{\\lambda \\supset \\mu }}{s}_{\\lambda }\n\]\n\nWe work in \( {\\Lambda }^{\\left( n\\right) } \) for any sufficiently large \( n \) ; of course \( n = k \) is sufficient. Let \( \\Delta \) denote the denominator i... | Yes |
Theorem 40.1. Let \( \lambda \) be a partition of \( k \) and let \( \mu \) be a partition of \( k - 1 \) . The following are equivalent.\n\n(i) The representation \( {\rho }_{\lambda } \) occurs in the representation of \( {S}_{k} \) induced from the representation \( {S}_{\mu } \) of \( {S}_{k - 1} \subset {S}_{k} \)... | Proof. Statements (i) and (ii) are equivalent by Frobenius reciprocity. Noting that \( {S}_{1} \) is the trivial group, we have \( {S}_{k - 1} = {S}_{k - 1} \times {S}_{1} \) . By definition, \( {\mathbf{s}}_{\mu }{\mathbf{e}}_{1} \) is the character of \( {S}_{k} \) induced from the character \( {\mathbf{s}}_{\mu } \o... | Yes |
Corollary 40.1. If \( \rho \) is an irreducible representation of \( {S}_{k - 1} \), then the representation of \( {S}_{k} \) induced from \( \rho \) is multiplicity-free; and if \( \tau \) is an irreducible representation of \( {S}_{k} \) then the representation of \( {S}_{k - 1} \) restricted from \( \tau \) is multi... | Proof. This is an immediate consequence of the theorem. | No |
Theorem 40.2. If \( \lambda \) is a partition of \( k \), the degree of the irreducible representation \( {\rho }_{\lambda } \) of \( {S}_{k} \) associated with \( \lambda \) is equal to the number of standard tableaux of shape \( \lambda \) . | Proof. Removing the top box (labeled \( k \) ) from a tableau of shape \( \lambda \) results in another tableau, of shape \( \mu \) (say), where \( \mu \subset \lambda \) . Thus, the set of tableaux of shape \( \lambda \) is in bijection with the set of tableaux of shape \( \mu \), where \( \mu \) runs through the part... | Yes |
Theorem 40.3. (Hook length formula) Let \( \lambda \) be a partition of \( k \) . The number of standard tableaux of shape \( \lambda \) equals \( k \) ! divided by the product of the lengths of the hooks. | Proof. See Exercise 40.5. | No |
Proposition 40.2. The coefficients \( {c}_{\lambda \mu }^{\nu } \) are nonnegative integers. | Proof. This is clear from any one of the characterizations in Theorem 38.3. | No |
Theorem 40.4. (Pieri’s formula) Let \( \mu \) be a partition of \( k \), and let \( r \geq 0 \) . Then \( {\mathbf{s}}_{\mu }{\mathbf{e}}_{r} \) is the sum of the \( {\mathbf{s}}_{\lambda } \) as \( \lambda \) runs through the partitions of \( k + r \) containing \( \mu \) such that \( \lambda \smallsetminus \mu \) is ... | Proof. Since by Theorems 34.3 and 35.2 applying the involution \( \iota \) interchanges \( {\mathbf{e}}_{r} \) and \( {\mathbf{h}}_{r} \) and also interchanges \( {\mathbf{s}}_{\mu } \) and \( {\mathbf{s}}_{\lambda } \), the second statement follows from the first, which we prove.\n\nThe proof that \( {\mathbf{s}}_{\mu... | Yes |
Proposition 41.1. Suppose that \( {\lambda }_{n} \) and \( {\mu }_{n - 1} \) are nonnegative, so the integer sequences \( \lambda \) and \( \mu \) are partitions. Then \( \lambda \) and \( \mu \) interlace if and only if \( \lambda \supset \mu \) and the skew partition \( \lambda \smallsetminus \mu \) is a horizontal s... | Proof. Assume that \( \lambda \supset \mu \) and \( \lambda \smallsetminus \mu \) is a horizontal strip. Then \( {\lambda }_{j} \geq {\mu }_{j} \) because \( \lambda \supset \mu \) . We must show that \( {\mu }_{j} \geq {\lambda }_{j + 1} \) . If it is not, \( {\lambda }_{j} \geq {\lambda }_{j + 1} > {\mu }_{j} \), whi... | Yes |
Theorem 41.1. Let \( \lambda = \left( {{\lambda }_{1},\ldots ,{\lambda }_{n}}\right) \) and \( \mu = \left( {{\mu }_{1},\ldots ,{\mu }_{n - 1}}\right) \) be integer sequences with \( {\lambda }_{1} \geq {\lambda }_{2} \geq \cdots \) and \( {\mu }_{1} \geq {\mu }_{2} \geq \cdots \) . Then the restriction of \( {\pi }_{\... | Proof. We restriction the representation \( {\pi }_{\lambda } \) of \( {\mathrm{{GL}}}_{n}\left( \mathbb{C}\right) \) to \( {\mathrm{{GL}}}_{n - 1}\left( \mathbb{C}\right) \) in two stages. First, we restrict it to \( {\mathrm{{GL}}}_{n - 1}\left( \mathbb{C}\right) \times {\mathrm{{GL}}}_{1}\left( \mathbb{C}\right) \),... | Yes |
Theorem 41.2. The degree of the irreducible representation \( {\pi }_{\lambda } \) of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) equals the number of Gelfand-Tsetlin patterns whose top row is \( \lambda \) . | Proof. The proof is identical in structure to Theorem 40.2. The Gelfand-Tsetlin patterns of shape \( \lambda \) can be counted by noting that striking the top row gives a Gelfand-Tsetlin pattern with a top row that is a partition \( \mu \) of length \( n - 1 \) that interlaces with \( \lambda \) . By induction, the num... | Yes |
Proposition 41.2. Let \( \lambda \) be a partition of length \( \leq n \) . The degree of the irreducible representation \( {\pi }_{\lambda } \) of \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) equals the number semistandard Young tableaux of shape \( \lambda \) with entries in \( \{ 1,2,\ldots, n\} \) . | Proof. In view of Theorem 41.2 it is sufficient to exhibit a bijection between these tableaux and the Gelfand-Tsetlin patterns with top row \( \lambda \) . We will explain how to go from the tableau to the Gelfand-Tsetlin pattern. Given a tableau, the top row of the Gelfand-Tsetlin pattern is the shape: of the tableau:... | No |
Theorem 42.1 (Heine, Szegö, Bump, Diaconis). Let \( f \in {L}^{1}\left( \mathbb{T}\right) \) be given, with \( f\left( t\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{d}_{n}{t}^{n} \) . Let \( \lambda \) and \( \mu \) be partitions of length \( \leq n \) . Define a function \( {\Phi }_{n, f} \) on \( \math... | Proof. By the Weyl integration formula in the form (22.18), and the Weyl character formula in the form (22.17), we have \[ {\int }_{\mathrm{U}\left( n\right) }{\Phi }_{n, f}\left( g\right) \overline{{\chi }_{\lambda }\left( g\right) }{\chi }_{\mu }\left( g\right) \mathrm{d}g \] \[ = \frac{1}{n!}{\int }_{\mathbb{T}}{\Ph... | Yes |
The irreducible complex representation \( \pi \) is self-contragredient if and only if there exists a nondegenerate bilinear form \( B : V \times V \rightarrow \) \( \mathbb{C} \) such that\n\n\[ B\left( {\pi \left( g\right) v,\pi \left( g\right) w}\right) = B\left( {v, w}\right) . \]\n\n(43.1)\n\nThe form \( B \) is u... | Proof. To emphasize the symmetry between \( V \) and \( {V}^{ * } \), let us write the dual pairing \( V \times {V}^{ * } \rightarrow \mathbb{C} \) in the symmetrical form \( L\left( v\right) = \llbracket v, L\rrbracket \) . The contragredient representation thus satisfies \( \llbracket \pi \left( g\right) v, L\rrbrack... | Yes |
Theorem 43.1 (Frobenius and Schur). Let \( \\left( {\\pi, V}\\right) \) be an irreducible representation of the compact group \( G \) . Then\n\n\[ \n{\\epsilon }_{\\pi } = {\\int }_{G}\\chi \\left( {g}^{2}\\right) \\mathrm{d}g \n\] | Proof. We have \( {p}_{2} = {h}_{2} - {e}_{2} \) in \( {\\Lambda }^{\\left( n\\right) } \) . Indeed, \( {p}_{2}\\left( {{x}_{1},\\ldots ,{x}_{n}}\\right) \) equals\n\n\[ \n\\mathop{\\sum }\\limits_{i}{x}_{i}^{2} = \\left( {\\mathop{\\sum }\\limits_{i}{x}_{i}^{2} + \\mathop{\\sum }\\limits_{{i < j}}{x}_{i}{x}_{j}}\\righ... | Yes |
Theorem 43.2. Let \( G \) be a finite group. Let \( \mu : G \rightarrow \mathbb{C} \) be the sum of the irreducible characters of \( G \) .\n\n(i) Suppose that \( {\epsilon }_{\pi } = 1 \) for every irreducible representation \( \pi \) . Then, for any \( g \in G,\mu \left( g\right) \) is the number of solutions to the ... | Proof. If \( \pi \) is an irreducible representation of \( G \), let \( {\chi }_{\pi } \) be its character. We will show\n\n\[ \mathop{\sum }\limits_{{\text{irreducible }\pi }}{\chi }_{\pi }\left( g\right) {\epsilon }_{\widehat{\pi }} = \# \left\{ {x \in G \mid {x}^{2} = g}\right\} . \]\n\n(43.2)\n\nIndeed, by Theorem ... | Yes |
Theorem 43.3. Every irreducible representation of \( {S}_{k} \) is defined over \( \mathbb{Q} \) . | Proof. The construction of Theorem 35.1 contained no reference to the ground field and works just as well over \( \mathbb{Q} \) . Specifically, our formulation of Mackey theory was valid over an arbitrary field, so if \( \lambda \) and \( \mu \) are conjugate partitions, the computation of Proposition 35.5 shows that t... | Yes |
Proposition 43.3. The restriction of \( {\omega }_{2r} \) to \( {S}_{{2r} - 1} \) is isomorphic to the character of \( {S}_{{2r} - 1} \) induced from the character \( {\omega }_{{2r} - 2} \) to \( {S}_{{2r} - 1} \) . | Proof. First, let us show that \( {B}_{2r} \smallsetminus {S}_{2r}/{S}_{{2r} - 1} \) consists of a single double coset. Indeed, \( {S}_{2r} \) acts transitively on \( X = \{ 1,2,\ldots ,{2r}\} \), and the stabilizer of \( {2r} \) is \( {S}_{{2r} - 1} \) . Therefore, we can identify \( {S}_{2r}/{S}_{{2r} - 1} \) with \(... | Yes |
Lemma 43.1. Every partition of \( {2r} - 1 \) having exactly one odd part is contained in a unique even partition of \( {2r} \) . | Proof. Let \( \mu \) be a partition of \( {2r} - 1 \) having exactly one odd part \( {\mu }_{i} \) . The unique even partition of \( {2r} \) containing \( \mu \) is \( {R}_{i}\mu \) . Note that this is a partition since \( i = 1 \) or \( {\mu }_{i} < {\mu }_{i - 1} \) . (We cannot have \( {\mu }_{i} \) and \( {\mu }_{i... | No |
Theorem 43.4. The character \( {\omega }_{2r} \) of \( {S}_{2r} \) is multiplicity-free. It is the sum of all irreducible characters \( {\mathbf{s}}_{\lambda } \) with \( \lambda \) an even partition of \( {2r} \) . | Proof. By induction, we may assume that this is true for \( {S}_{{2r} - 2} \) . The restriction of \( {\omega }_{2r} \) to \( {S}_{{2r} - 1} \) is the same as the character induced from \( {w}_{{2r} - 2} \) by Proposition 43.3. Using the branching rule for the symmetric groups, its irreducible constituents consist of a... | Yes |
Theorem 43.5 (Klyachko, Inglis, Richardson, and Saxl). Every irreducible character of \( {S}_{k} \) occurs with multiplicity 1 in the sum\n\n\[ \n{\bigoplus }_{{2r} \leq k}{\operatorname{Ind}}_{{B}_{2r} \times {S}_{k - {2r}}}^{{S}_{k}}\left( {1 \otimes \varepsilon }\right)\n\]\n\nwhere \( \varepsilon \) is the alternat... | Proof. We will show that \( {\operatorname{Ind}}_{{B}_{2r} \times {S}_{k - {2r}}}^{{S}_{k}}\left( {1 \otimes \varepsilon }\right) \) is the sum of the \( {\mathbf{s}}_{\lambda } \) as \( \lambda \) runs through the partitions of \( k \) having exactly \( k - {2r} \) odd parts. Indeed, it is obvious that if \( \lambda \... | Yes |
Proposition 44.1. Suppose that \( \rho : G \rightarrow \mathrm{{GL}}\left( V\right) \) is a representation and \( {\gamma }_{1},\ldots ,{\gamma }_{d} \) are the eigenvalues of \( \rho \left( g\right) \) . Then\n\n\[ \n{P}_{\rho }^{ \vee }\left( {g, t}\right) = \mathop{\prod }\limits_{i}{\left( 1 - t{\gamma }_{i}\right)... | Proof. The traces of \( \rho \left( g\right) \) on \( { \vee }^{k}V \) and \( { \land }^{k}V \) are\n\n\[ \n{h}_{k}\left( {{\gamma }_{1},\ldots ,{\gamma }_{d}}\right) \;\text{ and }\;{e}_{k}\left( {{\gamma }_{1},\ldots ,{\gamma }_{d}}\right) ,\n\]\n\nso this is a restatement of (33.1) and (33.2).\n\nWe see that for all... | Yes |
Theorem 44.1. Let \( V = {\mathbb{C}}^{n} \) be regarded as a \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) -module in the usual way. Then\n\n\[ \n{ \vee }^{k}\left( {{ \vee }^{2}V}\right) \cong {\bigoplus }_{\lambda \text{ an even partition of }{2k}}{\pi }_{\lambda }\n\] | Proof. This follows from Proposition 44.2, Theorem 36.4, and the explicit decomposition of Theorem 43.4. | No |
Theorem 44.2 (D. E. Littlewood). Let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be complex numbers, \( \left| {\alpha }_{i}\right| < 1 \) . Then\n\n\[ \mathop{\prod }\limits_{{1 \leq i \leq j \leq n}}{\left( 1 - {\alpha }_{i}{\alpha }_{j}\right) }^{-1} = \mathop{\sum }\limits_{{\lambda \text{ even }}}{s}_{\lambda }\left... | Proof. This follows on applying (44.2) to the symmetric square representation by using Proposition 44.1 and the explicit decomposition of Theorem 44.2. | No |
Theorem 44.3 (D. E. Littlewood). Let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be complex numbers, \( \left| {\alpha }_{i}\right| < 1 \) . Then\n\n\[ \left\lbrack {\mathop{\prod }\limits_{{1 \leq i \leq n}}\left( {1 + {\alpha }_{i}}\right) }\right\rbrack \left\lbrack {\mathop{\prod }\limits_{{1 \leq i \leq j \leq n}}{\... | Proof. The coefficient of \( {t}^{k} \) in\n\n\[ \left\lbrack {\mathop{\prod }\limits_{{1 \leq i \leq n}}\left( {1 + t{\alpha }_{i}}\right) }\right\rbrack \left\lbrack {\mathop{\prod }\limits_{{1 \leq i \leq j \leq n}}{\left( 1 - {t}^{2}{\alpha }_{i}{\alpha }_{j}\right) }^{-1}}\right\rbrack \]\n\n\[ = \left\lbrack {\ma... | Yes |
Proposition 44.3. The character \( {\widetilde{\omega }}_{2k} \) is the sum of the \( {\mathbf{s}}_{\lambda } \), where \( \lambda \) runs through all the partitions of \( k \) such that the conjugate partition \( {\lambda }^{t} \) is even. | Proof. This may be deduced from Theorem 43.4 as follows. Applying this with \( G = {S}_{2k}, H = {B}_{2k} \), and \( \rho = \varepsilon \), we see that \( {\widetilde{\omega }}_{2k} \) is the same as \( {\omega }_{2k} \) multiplied by the character \( \varepsilon \) . By Theorem 37.4, this is ’ \( {\omega }_{2k} \), an... | Yes |
Theorem 44.4. Let \( V = {\mathbb{C}}^{n} \) be regarded as a \( \mathrm{{GL}}\left( {n,\mathbb{C}}\right) \) -module in the usual way. Then\n\n\[ \n{ \vee }^{k}\left( {{ \land }^{2}V}\right) \cong \left( {{\bigotimes }^{2k}V}\right) { \otimes }_{\mathbb{C}\left\lbrack {S}_{2k}\right\rbrack }{\widetilde{\omega }}_{2k}\... | Proof. Similar to Theorem 44.2. | No |
Theorem 44.5 (D. E. Littlewood). Let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be complex numbers, \( \left| {\alpha }_{i}\right| < 1 \) . Then\n\n\[ \mathop{\prod }\limits_{{1 \leq i < j \leq n}}{\left( 1 - {\alpha }_{i}{\alpha }_{j}\right) }^{-1} = \mathop{\sum }\limits_{{{\lambda }^{t}\text{ even }}}{s}_{\lambda }\l... | Proof. Similar to Theorem 44.2. | No |
Theorem 44.6 (D. E. Littlewood). Let \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) be complex numbers, \( \left| {\alpha }_{i}\right| < 1 \) . Then\n\n\[ \left\lbrack {\mathop{\prod }\limits_{{1 \leq i \leq n}}{\left( 1 - {\alpha }_{i}\right) }^{-1}}\right\rbrack \left\lbrack {\mathop{\prod }\limits_{{1 \leq i < j \leq n}}... | Proof. Similar to Theorem 44.3, and actually equivalent to Theorem 44.3 using the identity \( \left( {1 + {\alpha }_{i}}\right) {\left( 1 - {\alpha }_{i}^{2}\right) }^{-1} = {\left( 1 - {\alpha }_{i}\right) }^{-1} \) . | No |
Proposition 45.1. Suppose \( \theta \) is a representation of a finite group \( G \) . A necessary and sufficient condition that \( \theta \) be multiplicity-free is that the ring \( {\operatorname{End}}_{G}\left( \theta \right) \) be commutative. | Proof. In the decomposition (45.1), we have \( {\operatorname{End}}_{G}\left( \theta \right) = \bigoplus {\operatorname{Mat}}_{{d}_{i}}\left( \mathbb{C}\right) \) . This is commutative if and only if all \( {d}_{i} \leq 1 \) . | Yes |
Proposition 45.2. Let \( G \) be a finite group, and let \( {H}_{1},{H}_{2},{H}_{3} \) be subgroups. Let \( \left( {{\pi }_{i},{V}_{i}}\right) \) be complex representations of \( {H}_{1},{H}_{2} \), and \( {H}_{3} \) and let \( {L}_{1} \) : \( {V}_{1}^{G} \rightarrow {V}_{2}^{G} \) and \( {L}_{2} : {V}_{2}^{G} \rightar... | Proof. Note that, using (32.9), the summand \( {\Delta }_{2}\left( {g{\gamma }^{-1}}\right) {\Delta }_{1}\left( \gamma \right) \) does not depend on the choice of representative \( \gamma \in {H}_{2} \smallsetminus G \) . The result is easily checked. | No |
Theorem 45.1. Let \( H \) be a subgroup of the finite group \( G \), and let \( \left( {\pi, V}\right) \) be a representation of \( H \) . Then \( \left( {G, H,\pi }\right) \) is a Gelfand triple if and only if the convolution algebra \( \mathcal{H} \) of functions \( \Delta : G \rightarrow {\operatorname{End}}_{\mathb... | Proof. By Proposition 45.2, this condition is equivalent to the commutativity of the endomorphism ring \( {\operatorname{End}}_{G}\left( {V}^{G}\right) \), so this follows from Proposition 45.1. | No |
Theorem 45.2. Let \( H \) be a subgroup of the finite group \( G \), and suppose that \( G \) admits an involution fixing \( H \) such that every double coset of \( H \) is invariant: \( {HgH} = {H}^{\iota }{gH} \). Then \( H \) is a Gelfand subgroup. | Proof. The ring \( \mathcal{H} \) of Theorem 45.1 is just the convolution ring of \( H \) -bi-invariant functions on \( G \). We have an involution on this ring:\n\n\[ \n{}^{\iota }\Delta \left( g\right) = \Delta \left( {{}^{\iota }g}\right) \n\]\n\nIt is easy to check that\n\n\[ \n{}^{\iota }\left( {{\Delta }_{1} * {\... | Yes |
Proposition 45.3. The subgroup \( {S}_{n} \times {S}_{m} \) is a Gelfand subgroup of \( {S}_{n + m} \) . | Proof. Let \( H = {S}_{n} \times {S}_{m} \) and \( G = {S}_{n + m} \) . We take the involution \( \iota \) in Theorem 45.2 to be the inverse map \( g \rightarrow {g}^{-1} \) . We must check that each double coset is \( \iota \) -stable.\n\nIt will be convenient to represent elements of \( {S}_{n + m} \) by permutation ... | Yes |
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