Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
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Theorem 9. The following properties are equivalent:\n\n(i) \( \mathrm{G} \) is abelian.\n\n(ii) All the irreducible representations of \( \mathrm{G} \) have degree 1 . | Let \( g \) be the order of \( \mathrm{G} \), and let \( \left( {{n}_{1},\ldots ,{n}_{h}}\right) \) be the degrees of the distinct irreducible representations of \( \mathrm{G} \) ; we know, cf. Ch. 2, that \( h \) is the number of classes of \( \mathrm{G} \), and that \( g = {n}_{1}^{2} + \cdots + {n}_{h}^{2} \) . Henc... | Yes |
(i) If \( {\rho }^{1} \) and \( {\rho }^{2} \) are irreducible, \( {\rho }^{1} \otimes {\rho }^{2} \) is an irreducible representation of \( {\mathrm{G}}_{1} \times {\mathrm{G}}_{2} \) . | If \( {\rho }^{1} \) and \( {\rho }^{2} \) are irreducible, we have (cf. 2.3):\n\n\[ \frac{1}{{g}_{1}}\mathop{\sum }\limits_{{s}_{1}}{\left| {\chi }_{1}\left( {s}_{1}\right) \right| }^{2} = 1,\;\frac{1}{{g}_{2}}\mathop{\sum }\limits_{{s}_{2}}{\left| {\chi }_{2}\left( {s}_{2}\right) \right| }^{2} = 1. \]\n\nBy multiplic... | Yes |
Lemma 1. Suppose that \( \left( {\mathrm{V},\rho }\right) \) is induced by \( \left( {\mathrm{W},\theta }\right) \) . Let \( {\rho }^{\prime } : \mathrm{G} \rightarrow \mathrm{{GL}}\left( {\mathrm{V}}^{\prime }\right) \) be a linear representation of \( \mathrm{G} \), and let \( f : \mathrm{W} \rightarrow {\mathrm{V}}^... | If \( \mathrm{F} \) satisfies these conditions, and if \( x \in {\rho }_{s}\mathrm{\;W} \), we have \( {\rho }_{s}^{-1}x \in \mathrm{W} \) ; hence\n\n\[ \mathrm{F}\left( x\right) = \mathrm{F}\left( {{\rho }_{s}{\rho }_{s}^{-1}x}\right) = {\rho }_{s}^{\prime }\mathrm{F}\left( {{\rho }_{s}^{-1}x}\right) = {\rho }_{s}^{\p... | Yes |
Theorem 11. Let \( \left( {\mathrm{W},\theta }\right) \) be a linear representation of \( \mathrm{H} \). There exists a linear representation \( \left( {\mathrm{V},\rho }\right) \) of \( \mathrm{G} \) which is induced by \( \left( {\mathrm{W},\theta }\right) \), and it is unique up to isomorphism. | Let us first prove the existence of the induced representation \( \rho \). In view of example 3, above, we may assume that \( \theta \) is irreducible. In this case, \( \theta \) is isomorphic to a subrepresentation of the regular representation of \( \mathrm{H} \), which can be induced to the regular representation of... | Yes |
Theorem 12. Let \( h \) be the order of \( \mathrm{H} \) and let \( \mathrm{R} \) be a system of representatives of \( \mathrm{G}/\mathrm{H} \) . For each \( u \in \mathrm{G} \), we have\n\n\[ \n{\chi }_{\rho }\left( u\right) = \mathop{\sum }\limits_{\substack{{r \in \mathrm{R}} \\ {{r}^{-1}{ur} \in \mathrm{H}} }}{\chi... | The space \( \mathrm{V} \) is the direct sum of the \( {\rho }_{r}\mathrm{\;W}, r \in \mathrm{R} \) . Moreover \( {\rho }_{u} \) permutes the \( {\rho }_{r}\mathrm{\;W} \) among themselves. More precisely, if we write \( {ur} \) in the form \( {r}_{u}t \) with \( {r}_{u} \in \mathrm{R} \) and \( t \in \mathrm{H} \), we... | Yes |
Proposition 9. If \( \mathrm{K} \) is a field of characteristic zero, the algebra \( \mathrm{K}\left\lbrack \mathrm{G}\right\rbrack \) is semisimple. | To say that \( \mathrm{K}\left\lbrack \mathrm{G}\right\rbrack \) is a semisimple algebra is equivalent to saying that each \( \mathrm{K}\left\lbrack \mathrm{G}\right\rbrack \) -module \( \mathrm{V} \) is semisimple, i.e., that each submodule \( {\mathrm{V}}^{\prime } \) of \( \mathrm{V} \) is a direct factor in \( \mat... | Yes |
Proposition 10. The homomorphism \( \widetilde{\rho } \) defined above is an isomorphism. | This is a general property of semisimple algebras. In the present case, it can be verified in the following way: First, \( \widetilde{\rho } \) is surjective. Otherwise there would exist a nonzero linear form on \( \prod {\mathbf{M}}_{{n}_{i}}\left( \mathbf{C}\right) \) vanishing on the image of \( \widetilde{\rho } \)... | Yes |
Proposition 11 (Fourier inversion formula). Let \( {\left( {u}_{i}\right) }_{1 \leq i \leq h} \) be an element of \( \prod \operatorname{End}\left( {\mathrm{W}}_{i}\right) \), and let \( u = \mathop{\sum }\limits_{{s \in \mathrm{G}}}u\left( s\right) s \) be the element of \( \mathbf{C}\left\lbrack \mathrm{G}\right\rbra... | By linearity it is enough to check the formula when \( u \) is equal to an element \( t \) of \( \mathrm{G} \) . We have then\n\n\[ u\left( s\right) = {\delta }_{st}\;\text{ and }\;{\operatorname{Tr}}_{{\mathrm{W}}_{i}}\left( {{\rho }_{i}\left( {s}^{-1}\right) {u}_{i}}\right) = {\chi }_{i}\left( {{s}^{-1}t}\right) ,\]\... | Yes |
Proposition 12. The homomorphism \( {\widetilde{\rho }}_{i} \) maps the center of \( \mathbf{C}\left\lbrack \mathrm{G}\right\rbrack \) into the set of homotheties of \( {\mathrm{W}}_{i} \) and defines an algebra homomorphism\n\n\[ \n{\omega }_{i} : \text{ Cent. }\mathbf{C}\left\lbrack \mathrm{G}\right\rbrack \rightarro... | This is just a reformulation of prop. 6 of 2.5. | No |
Proposition 13. The family \( {\left( {\omega }_{i}\right) }_{1 \leq i \leq h} \) defines an isomorphism of Cent. C[G] onto the algebra \( {\mathbf{C}}^{h} = \mathbf{C} \times \cdots \times \mathbf{C} \) . | If we identify \( \mathbf{C}\left\lbrack \mathrm{G}\right\rbrack \) with the product of the \( \operatorname{End}\left( {\mathrm{W}}_{i}\right) \), the center of \( \mathbf{C}\left\lbrack \mathrm{G}\right\rbrack \) becomes the product of the centers of the \( \operatorname{End}\left( {\mathrm{W}}_{i}\right) \) . But th... | Yes |
Proposition 14. Let \( x \) be an element of a commutative ring \( \mathrm{R} \) . The following properties are equivalent:\n\n(i) \( x \) is integral over \( \mathbf{Z} \) .\n\n(ii) The subring \( \mathbf{Z}\left\lbrack x\right\rbrack \) of \( \mathbf{R} \) generated by \( x \) is finitely generated as a \( \mathbf{Z}... | The equivalence of (ii) and (iii) follows from the fact that a submodule of a finitely generated \( \mathbf{Z} \) -module is finitely generated, since \( \mathbf{Z} \) is noetherian. On the other hand, if \( x \) satisfies an equation\n\n\[ {x}^{n} + {a}_{1}{x}^{n - 1} + \cdots + {a}_{n} = 0,\;\text{ with }{a}_{i} \in ... | Yes |
Corollary 1. If \( \mathrm{R} \) is a finitely generated \( \mathbf{Z} \) -module, each element of \( \mathrm{R} \) is integral over \( \mathbf{Z} \) . | This follows from the implication (iii) \( \Rightarrow \) (i). | No |
Corollary 2. The elements of \( \mathbf{R} \) which are integral over \( \mathbf{Z} \) form a subring of \( \mathbf{R} \). | Let \( x, y \in \mathbf{R} \) ; if \( x, y \) are integral over \( \mathbf{Z} \), the rings \( \mathbf{Z}\left\lbrack x\right\rbrack \) and \( \mathbf{Z}\left\lbrack y\right\rbrack \) are finitely generated over \( \mathbf{Z} \). The same is then true of their tensor product \( \mathbf{Z}\left\lbrack x\right\rbrack \ot... | No |
Proposition 15. Let \( \chi \) be the character of a representation \( \rho \) of a finite group G. Then \( \chi \left( s\right) \) is an algebraic integer for each \( s \in \mathbf{G} \) . | Indeed \( \chi \left( s\right) \) is the trace of \( \rho \left( s\right) \), hence is the sum of eigenvalues of \( \rho \left( s\right) \) , which are roots of unity. | Yes |
Proposition 16. Let \( u = \sum u\left( s\right) s \) be an element of Cent. C[G] such that the \( u\left( s\right) \) are algebraic integers. Then \( u \) is integral over \( \mathbf{Z} \) . | Let \( {c}_{i}\left( {1 \leq i \leq h}\right) \) be the conjugacy classes of \( \mathrm{G} \) and put \( {e}_{i} = \mathop{\sum }\limits_{{s \in {c}_{i}}}s \) , cf. 6.3. For \( {s}_{i} \in {c}_{i} \) we can write \( u \) in the form \( u = \mathop{\sum }\limits_{{i = 1}}^{{i = h}}u\left( {s}_{i}\right) {e}_{i} \) . In ... | Yes |
Corollary 1. Let \( \rho \) be an irreducible representation of \( \mathrm{G} \) of degree \( n \) and character \( \chi \) . If \( u \) is as above, then the number \( \left( {1/n}\right) \mathop{\sum }\limits_{{s \in G}}u\left( s\right) \chi \left( s\right) \) is an algebraic integer. | Indeed, this number is the image of \( u \) under the homomorphism\n\n\[\n\omega : \text{ Cent. }\mathbf{C}\left\lbrack \mathrm{G}\right\rbrack \rightarrow \mathbf{C}\n\]\n\nassociated with \( \rho \) (cf. prop. 12). As \( u \) is integral over \( \mathbf{Z} \), the same is true of its image under \( \omega \) . | Yes |
Corollary 2. The degrees of the irreducible representations of \( \mathrm{G} \) divide the order of \( \mathbf{G} \) . | Let \( g \) be the order of \( \mathrm{G} \) . We apply cor. 1 to the element \( u \) \( = \mathop{\sum }\limits_{{s \in \mathrm{G}}}\chi \left( {s}^{-1}\right) s \), which is legitimate since \( \chi \) is a class function and since the \( \chi \left( s\right) \) are algebraic integers (prop. 15); we obtain that the n... | Yes |
Proposition 17. Let \( \mathrm{C} \) be the center of \( \mathrm{G} \) . The degrees of the irreducible representations of \( \mathrm{G} \) divide \( \left( {\mathrm{G} : \mathrm{C}}\right) \) . | Let \( g \) be the order of \( \mathrm{G} \) and \( c \) that of \( \mathrm{C} \), and let \( \rho : \mathrm{G} \rightarrow \mathrm{{GL}}\left( \mathrm{W}\right) \) be an irreducible representation of \( \mathrm{G} \) of degree \( n \) . If \( s \in \mathrm{C},\rho \left( s\right) \) commutes with all the \( \rho \left... | Yes |
Proposition 18. In order that \( \mathrm{V} \) be induced by \( \mathrm{W} \), it is necessary and sufficient that the homomorphism\n\n\[ i : \mathbf{C}\left\lbrack \mathrm{G}\right\rbrack { \otimes }_{\mathbf{C}\left\lbrack \mathrm{H}\right\rbrack }\mathrm{W} \rightarrow \mathrm{V} \]\n\nbe an isomorphism. | This is a consequence of the fact that the elements of \( R \) form a basis of \( \mathbf{C}\left\lbrack \mathrm{G}\right\rbrack \) considered as a right \( \mathbf{C}\left\lbrack \mathrm{H}\right\rbrack \) -module. | No |
Proposition 19. Let \( \\mathrm{V} \) be a \( \\mathbf{C}\\left\\lbrack \\mathrm{G}\\right\\rbrack \) -module which is a direct sum \( \\mathrm{V} = { \\oplus }_{i \\in \\mathbf{I}}{\\mathrm{W}}_{i} \) of vector subspaces permuted transitively by \( \\mathrm{G} \) . Let \( {i}_{0} \\in \\mathrm{I},\\mathrm{W} = {\\math... | This is clear. | No |
Lemma 2. If \( {\varphi }_{1} \) and \( {\varphi }_{2} \) are the characters of \( {V}_{1} \) and \( {V}_{2} \), we have\n\n\[ \n{\left\langle {\varphi }_{1},{\varphi }_{2}\right\rangle }_{\mathrm{G}} = {\left\langle {\mathrm{V}}_{1},{\mathrm{\;V}}_{2}\right\rangle }_{\mathrm{G}} \n\] | Decomposing \( {V}_{1} \) and \( {V}_{2} \) into direct sums, we can assume that they are irreducible, in which case the lemma follows from the orthogonality formulas for characters (2.3, th. 3). | No |
Theorem 13 (Frobenius reciprocity). If \( \psi \) is a class function on \( \mathrm{H} \) and \( {\varphi a} \) class function on \( \mathrm{G} \), we have\n\n\[ \langle \psi ,\operatorname{Res}\varphi {\rangle }_{\mathrm{H}} = \langle \operatorname{Ind}\psi ,\varphi {\rangle }_{\mathrm{G}}. \] | Since each class function is a linear combination of characters, we can assume that \( \psi \) is the character of a \( \mathbf{C}\left\lbrack \mathrm{H}\right\rbrack \) -module \( \mathrm{W} \) and \( \varphi \) is the character of a C[G]-module E. In view of lemma 2, it is enough to show that\n\n(*) \n\n\[ \langle \m... | Yes |
Proposition 21. Let \( \mathrm{W} \) be an irreducible representation of \( \mathrm{H} \) and \( \mathrm{E} \) an irreducible representation of \( \mathrm{G} \) . Then the number of times that \( \mathrm{W} \) occurs in Res \( \mathrm{E} \) is equal to the number of times that \( E \) occurs in Ind \( \mathrm{W} \) | This follows from th. 13, applied to the character \( \psi \) of \( \mathrm{W} \) and to the character \( \varphi \) of \( \mathrm{E} \) (one may also apply formula (*)). | Yes |
Proposition 22. The representation \( {\operatorname{Res}}_{\mathrm{K}}{\operatorname{Ind}}_{\mathrm{H}}^{\mathrm{G}}\left( \mathrm{W}\right) \) is isomorphic to the direct sum of the representations \( {\operatorname{Ind}}_{{\mathrm{H}}_{s}}^{\mathrm{K}}\left( {\mathrm{W}}_{s}\right) \), for \( s \in \mathrm{S} \simeq... | We know that \( \mathrm{V} \) is the direct sum of the images \( x\mathrm{\;W} \), for \( x \in \mathrm{G}/\mathrm{H} \) . Let \( s \in \mathrm{S} \) and let \( \mathrm{V}\left( s\right) \) be the subspace of \( \mathrm{V} \) generated by the images \( x\mathrm{\;W} \), for \( x \in \mathrm{K}s\mathrm{H} \) ; the space... | Yes |
Proposition 23. In order that the induced representation \( \mathrm{V} = {\operatorname{Ind}}_{\mathrm{H}}^{\mathrm{G}}\mathrm{W} \) be irreducible, it is necessary and sufficient that the following two conditions be satisfied:\n\n(a) \( \mathrm{W} \) is irreducible.\n\n(b) For each \( s \in \mathrm{G} - \mathrm{H} \) ... | In order that \( \mathrm{V} \) be irreducible, it is necessary and sufficient that \( \langle \mathrm{V},\mathrm{V}{\rangle }_{\mathrm{G}} = 1 \) . But, according to Frobenius reciprocity, we have:\n\n\[ \langle \mathrm{V},\mathrm{V}{\rangle }_{\mathrm{G}} = {\left\langle \mathrm{W},{\operatorname{Res}}_{\mathrm{H}}\ma... | Yes |
Proposition 24. Let \( \mathrm{A} \) be a normal subgroup of a group \( \mathrm{G} \), and let \( \rho : \mathrm{G} \rightarrow \mathrm{{GL}}\left( \mathrm{V}\right) \) be an irreducible representation of \( \mathrm{G} \) . Then:\n\n(a) either there exists a subgroup \( \mathrm{H} \) of \( \mathrm{G} \), unequal to \( ... | Let \( \mathrm{V} = \oplus {\mathrm{V}}_{i} \) be the canonical decomposition of the representation \( \rho \) (restricted to A) into a direct sum of isotypic representations (cf. 2.6). For \( s \in \mathrm{G} \) we see by \ | No |
(a) \( {\theta }_{i,\rho } \) is irreducible. | We prove (a) using Mackey's criterion (7.4, prop. 23) as follows: Let \( s \notin {\mathrm{G}}_{i} = \mathrm{A} \cdot {\mathrm{H}}_{i} \), and let \( {\mathrm{K}}_{s} = {\mathrm{G}}_{i} \cap s{\mathrm{G}}_{i}{s}^{-1} \) . We have to show that, if we compose the representation \( {\chi }_{i} \otimes \widetilde{\rho } \)... | Yes |
Theorem 14. Every p-group is nilpotent (thus supersolvable). | In view of the preceding it suffices to show that the center of every nontrivial \( p \) -group \( \mathrm{G} \) is nontrivial. This is a consequence of the following lemma: | No |
Lemma 3. Let \( \mathrm{G} \) be a p-group acting on a finite set \( \mathrm{X} \), and let \( {\mathrm{X}}^{\mathrm{G}} \) be the set of elements of \( \mathrm{X} \) fixed by \( \mathrm{G} \). We have\n\n\[ \operatorname{Card}\left( \mathrm{X}\right) \equiv \operatorname{Card}\left( {\mathrm{X}}^{\mathrm{G}}\right) \;... | Indeed \( \mathrm{X} - {\mathrm{X}}^{\mathrm{G}} \) is a union of nontrivial orbits of \( \mathrm{G} \), and the cardinality of each of these orbits is a power \( {p}^{\alpha } \) of \( p \), with \( \alpha \geq 1 \) ; hence \( \operatorname{Card}\left( {\mathrm{X} - {\mathrm{X}}^{\mathrm{G}}}\right) \) is divisible by... | Yes |
Proposition 26. Let \( \mathrm{V} \) be a vector space \( \neq 0 \) over a field \( k \) of characteristic \( p \) and let \( \rho : \mathrm{G} \rightarrow \mathrm{{GL}}\left( \mathrm{V}\right) \) be a linear representation of a p-group \( \mathrm{G} \) in \( \mathrm{V} \) . Then there exists a nonzero element of \( \m... | Let \( x \) be a nonzero element of \( \mathrm{V} \), and let \( \mathrm{X} \) be the subgroup of \( \mathrm{V} \) generated by the \( \rho \left( s\right) x, s \in \mathrm{G} \) . We apply lemma 3 to \( \mathrm{X} \), observing that \( \mathrm{X} \) is finite and of order a power of \( p \) . Therefore \( {\mathrm{X}}... | No |
There exist Sylow p-subgroups. | To prove (a) we use induction on the order of \( \mathrm{G} \) . We may assume \( n \geq 1 \) , i.e. Card \( \left( \mathrm{G}\right) \equiv 0\left( {\;\operatorname{mod}.p}\right) \) . Let \( \mathrm{C} \) be the center of \( \mathrm{G} \) . If Card \( \left( \mathrm{C}\right) \) is divisible of order \( p \), an elem... | Yes |
Lemma 4. Let \( \mathrm{G} \) be a nonabelian supersolvable group. Then there exists a normal abelian subgroup of \( \mathrm{G} \) which is not contained in the center of \( \mathrm{G} \) . | Let \( \mathrm{C} \) be the center of \( \mathrm{G} \) . The quotient \( \mathrm{H} = \mathrm{G}/\mathrm{C} \) is supersolvable, thus has a composition series in which the first nontrivial term \( {\mathrm{H}}_{1} \) is a cyclic normal subgroup of \( \mathrm{H} \) . The inverse image of \( {\mathrm{H}}_{1} \) in \( \ma... | Yes |
Theorem 16. Let \( \mathrm{G} \) be a supersolvable group. Then each irreducible representation of \( \mathrm{G} \) is induced by a representation of degree 1 of a subgroup of \( \mathrm{G} \) (i.e., is monomial). | We prove the theorem by induction on the order of G. Consequently we may consider only those irreducible representations \( \rho \) which are faithful, i.e., such that \( \operatorname{Ker}\left( \rho \right) = \{ 1\} \) . If \( \mathrm{G} \) is abelian, such a \( \rho \) is of degree 1 and there is nothing to prove. S... | Yes |
Theorem 17. Let \( \mathrm{X} \) be a family of subgroups of a finite group \( \mathrm{G} \) . Let Ind: \( { \oplus }_{\mathrm{H} \in \mathrm{X}}\mathrm{R}\left( \mathrm{H}\right) \rightarrow \mathrm{R}\left( \mathrm{G}\right) \) be the homomorphism defined by the family of \( {\operatorname{Ind}}_{\mathrm{H}}^{\mathrm... | First, we show that (ii) \( \Rightarrow \) (i). Let \( S \) be the union of the conjugates of the subgroups \( \mathrm{H} \) belonging to \( \mathrm{X} \) . Each function of the form \( \sum {\operatorname{Ind}}_{\mathrm{H}}^{\widetilde{\mathrm{G}}}\left( {f}_{\mathrm{H}}\right) \), with \( {f}_{\mathrm{H}} \in \mathrm... | Yes |
Proposition 27. If \( \mathrm{G} \) is a finite group of order \( g \), then\n\n\[ g = \mathop{\sum }\limits_{{\mathrm{A} \subset \mathrm{G}}}{\operatorname{Ind}}_{\mathrm{A}}^{\mathrm{G}}\left( {\theta }_{\mathrm{A}}\right) \]\n\nwhere \( \mathrm{A} \) runs through all the cyclic subgroups of \( \mathrm{G} \) . | Put \( {\theta }_{\mathrm{A}}^{\prime } = {\operatorname{Ind}}_{\mathrm{A}}^{\mathrm{G}}\left( {\theta }_{\mathrm{A}}\right) \) . For \( x \in \mathrm{G} \) we have\n\n\[ {\theta }_{\mathrm{A}}^{\prime }\left( x\right) = \frac{1}{a}\mathop{\sum }\limits_{\substack{{y \in \mathrm{G}} \\ {{yx}{y}^{-1} \in \mathrm{A}} }}{... | Yes |
Proposition 28. If \( \mathrm{A} \) is a cyclic group, then \( {\theta }_{\mathrm{A}} \in \mathrm{R}\left( \mathrm{A}\right) \) . | The proof is by induction on the order \( a \) of \( \mathrm{A} \), the case \( a = 1 \) being trivial. By prop. 27 we have\n\n\[ a = \mathop{\sum }\limits_{{\mathrm{B} \subset \mathrm{A}}}{\operatorname{Ind}}_{\mathrm{B}}^{\mathrm{A}}\left( {\theta }_{\mathrm{B}}\right) = {\theta }_{\mathrm{A}} + \mathop{\sum }\limits... | Yes |
Theorem 18. Let \( \mathrm{G} \) be a finite group and let \( {\mathrm{V}}_{p} \) be the subgroup of \( \mathrm{R}\left( \mathrm{G}\right) \) generated by characters induced from those of p-elementary subgroups of \( \mathbf{G} \) . Then the index of \( {\mathrm{V}}_{p} \) in \( \mathrm{R}\left( \mathrm{G}\right) \) is... | Let \( \mathrm{X}\left( p\right) \) be the family of \( p \) -elementary subgroups of \( \mathrm{G} \) . The group \( {\mathrm{V}}_{p} \) is the image of the homomorphism\n\n\[ \text{Ind:}\underset{\mathrm{H} \in \mathrm{X}\left( p\right) }{ \oplus }\mathrm{R}\left( \mathrm{H}\right) \rightarrow \mathrm{R}\left( \mathr... | Yes |
Lemma 6. Each class function on \( \mathrm{G} \) with integer values divisible by \( g \) is an A-linear combination of characters induced from characters of cyclic subgroups of \( \mathrm{G} \) . | Let \( f \) be such a function, and write it in the form \( {g\chi } \), where \( \chi \) is a class function with integer values. If \( \mathrm{C} \) is a cyclic subgroup of \( \mathrm{G} \), let \( {\theta }_{\mathrm{C}} \) be the element of \( \mathrm{R}\left( \mathrm{C}\right) \) defined in 9.4. We have\n\n\[ g = \... | Yes |
Lemma 7. Let \( \chi \) be an element of \( \mathrm{A} \otimes \mathrm{R}\left( \mathrm{G}\right) \) with integer values, let \( x \in \mathrm{G} \) , and let \( {x}_{r} \) be the \( {p}^{\prime } \) -component of \( x \) (cf. 10.1). Then\n\n\[ \chi \left( x\right) \equiv \chi \left( {x}_{r}\right) \left( {\;\operatorn... | By restriction, we are led to the case where \( \mathrm{G} \) is cyclic and generated by \( x \) . Now \( \chi = \sum {a}_{i}{\chi }_{i} \), with \( {a}_{i} \in \mathrm{A} \) and the \( {\chi }_{i} \) running over the distinct characters of degree 1 of \( \mathrm{G} \) . If \( q \) is a sufficiently large power of \( p... | Yes |
Lemma 8. Let \( x \) be a \( {p}^{\prime } \) -element of \( \mathrm{G} \), and let \( \mathrm{H} \) be a p-elementary subgroup of \( \mathrm{G} \) associated with \( x\left( {10.1}\right) \) . Then there exists a function \( \psi \in \mathrm{A} \otimes \mathrm{R}\left( \mathrm{H}\right) \) , with integer values, such ... | Let \( \mathrm{C} \) be the cyclic subgroup of \( \mathrm{G} \) generated by \( x \), and let \( \mathrm{Z}\left( x\right) \) be the centralizer of \( x \) in \( \mathrm{G} \) . We have \( \mathrm{H} = \mathrm{C} \times \mathrm{P} \), where \( \mathrm{P} \) is a Sylow \( p \) -subgroup of \( \mathrm{Z}\left( x\right) \... | Yes |
Lemma 9. There exists an element \( \psi \) of \( \mathrm{A} \otimes {\mathrm{V}}_{p} \), with integer values, such that \( \psi \left( x\right) ≢ 0\left( {{\;\operatorname{mod}\;.}p}\right) \) for each \( x \in \mathrm{G}. \) | Let \( {\left( {x}_{i}\right) }_{i \in I} \) be a system of representatives of the \( p \) -regular classes (i.e. those consisting of \( {p}^{\prime } \) -elements). Lemma 8 gives us an element \( {\psi }_{i} \) of \( \mathrm{A} \otimes {\mathrm{V}}_{p} \) , with integer values, such that\n\n\[ \n{\psi }_{i}\left( {x}_... | Yes |
Theorem 19. Each character of \( \mathrm{G} \) is a linear combination with integer coefficients of characters induced from characters of elementary subgroups. | Let \( {V}_{p} \) be the subgroup of \( R\left( G\right) \) defined in th. 18. It suffices to show that the sum \( \mathrm{V} \) of the \( {\mathrm{V}}_{p} \), for \( p \) prime, is equal to \( \mathrm{R}\left( \mathrm{G}\right) \) . Now \( \mathrm{V} \) contains \( {\mathrm{V}}_{p} \), so the index of \( \mathrm{V} \)... | Yes |
Theorem 20. Each character of \( \mathrm{G} \) is a linear combination with integer coefficients of monomial characters. | This follows from th. 19 and the fact that each character of an elementary group is monomial, since such a group is nilpotent (cf. 8.5, th. 16). | No |
Theorem 21. Let \( \varphi \) be a class function on \( \mathrm{G} \) such that, for each elementary subgroup \( \mathrm{H} \) of \( \mathrm{G} \), we have \( {\operatorname{Res}}_{\mathrm{H}}^{\mathrm{G}}\varphi \in \mathrm{B} \otimes \mathrm{R}\left( \mathrm{H}\right) \) . Then \( \varphi \in \mathrm{B} \otimes \math... | Let \( \mathrm{X} \) be the set of all elementary subgroups of \( \mathrm{G} \) . By th. 11, we can write the constant function 1 in the form\n\n\[ 1 = \mathop{\sum }\limits_{{\mathrm{H} \in \mathrm{X}}}{\operatorname{Ind}}_{\mathrm{H}}^{\mathrm{G}}{f}_{\mathrm{H}},\;\text{ with }{f}_{\mathrm{H}} \in \mathrm{R}\left( \... | Yes |
Theorem 22. Let \( \varphi \) be a class function on \( \mathrm{G} \) such that, for each elementary subgroup \( \mathrm{H} \) of \( \mathrm{G} \), and each character \( \chi \) of degree 1 of \( \mathrm{H} \), the number \[ {\left\langle \chi ,{\operatorname{Res}}_{\mathrm{H}}\varphi \right\rangle }_{\mathrm{H}} = \fr... | Let \( \mathrm{H} \) be an elementary subgroup of \( \mathrm{G} \) . Let \[ {\operatorname{Res}}_{\mathrm{H}}^{\mathrm{G}}\varphi = \mathop{\sum }\limits_{\omega }{c}_{\omega }\omega ,\;\text{ where }{c}_{\omega } = {\left\langle \omega ,{\operatorname{Res}}_{\mathrm{H}}\varphi \right\rangle }_{\mathrm{H}}, \] be the d... | Yes |
Proposition 29. The homomorphism Res: \( \mathrm{R}\left( \mathrm{G}\right) \rightarrow \underset{\mathrm{H} \in \mathrm{X}}{ \oplus }\mathrm{R}\left( \mathrm{H}\right) \) is a split injection. | It is immediate that Res is an injection. To show that it is split, it suffices to prove that its cokernel is torsion free, since the groups under consideration are finitely generated free \( \mathbf{Z} \) -modules. So we must show that, if \( f = {\left( {f}_{\mathrm{H}}\right) }_{\mathrm{H} \in \mathrm{X}} \) is an e... | Yes |
Theorem 23. If \( f \) is a class function on \( \mathrm{G} \) with values in \( \mathrm{A} \), the function \( \left( {g/\left( {g, n}\right) }\right) {\Psi }^{n}f \) belongs to \( \mathrm{A} \otimes \mathrm{R}\left( \mathrm{G}\right) \) . | If \( c \) is a conjugacy class of \( \mathrm{G} \), denote by \( {f}_{c} \) the characteristic function of \( c \) , which takes the value 1 on \( c \) and 0 on \( \mathrm{G} - c \) . The function \( {\Psi }^{n}{f}_{c} \) is given by:\n\n\[ \n{\Psi }^{n}{f}_{c}\left( x\right) = \left\{ \begin{array}{ll} 1 & \text{ if ... | Yes |
Lemma 10. Let \( c \) be a conjugacy class of a p-group \( \mathrm{G} \), let \( \chi \) be a character of degree 1 of \( \mathrm{G} \), and let \( {a}_{c} = \mathop{\sum }\limits_{{{x}^{n} \in c}}\chi \left( x\right) \) . Then \( {a}_{c} \equiv 0\left( {{\;\operatorname{mod}\;.}\left( {g, n}\right) \mathrm{A}}\right) ... | First, observe that the sum of the \( {a}_{c} \) (for \( \chi \) fixed and \( c \) variable) is equal to \( \mathop{\sum }\limits_{{x \in \mathrm{G}}}\chi \left( x\right) \), i.e., to \( g \) if \( \chi = 1 \) and to 0 otherwise. So\n\n\[ \mathop{\sum }\limits_{c}{a}_{c} \equiv 0\left( {{\;\operatorname{mod}\;.}\left( ... | Yes |
Lemma 11. Let \( p \) be a prime number. Let \( x \) be a \( {p}^{\prime } \) -element of \( \mathrm{G},\mathrm{C} \) the subgroup generated by \( x \), and \( \mathrm{P} \) a Sylow p-subgroup of the centralizer \( \mathbf{Z}\left( x\right) \) of \( x \) in \( \mathrm{G} \) . Let \( \mathrm{H} \) be a subgroup of \( \m... | Let \( \mathrm{S}\left( x\right) \) be the set of conjugates of \( x \) . Then\n\n\[ \n{\psi }^{\prime }\left( x\right) = \frac{\operatorname{Card}Z\left( x\right) }{\operatorname{Card}\mathrm{H}}\mathop{\sum }\limits_{{y \in \mathrm{S}\left( x\right) \cap \mathrm{H}}}\psi \left( y\right) \n\]\n\nLet \( {\left( {\mathr... | Yes |
Proposition 30. If\n\n(i) with each class \( c \in \mathrm{{Cl}}\left( \mathrm{G}\right) \) we associate \( {\mathrm{P}}_{0, c} \),\n\n(ii) with each p-regular class \( c \) and each maximal ideal \( \mathbf{M} \) of \( \mathbf{A} \) with residual characteristic \( p \) we associate \( {\mathrm{P}}_{\mathrm{M}, c} \),\... | Since \( \operatorname{Spec}\left( {\mathrm{A}}^{\mathrm{{Cl}}\left( \mathrm{G}\right) }\right) \rightarrow \operatorname{Spec}\left( {\mathrm{A} \otimes \mathrm{R}\left( \mathrm{G}\right) }\right) \) is surjective (cf. above), each prime ideal \( \mathfrak{p} \) of \( \mathrm{A} \otimes \mathrm{R}\left( \mathrm{G}\rig... | Yes |
Proposition 31. \( \operatorname{Spec}\left( {\mathrm{A} \otimes \mathrm{R}\left( \mathrm{G}\right) }\right) \) is connected in the Zariski topology. | Let \( x \) be an element of \( \mathrm{G} \) of order \( {p}_{1}^{{n}_{1}} \cdot {p}_{2}^{{n}_{2}}\cdots {p}_{k}^{{n}_{k}};x \) decomposes into a product \( x = {x}_{{p}_{1}} \cdot {x}_{{p}_{2}}\cdots {x}_{{p}_{k}} \), where \( {x}_{{p}_{i}} \) is of order \( {p}_{i}^{{n}_{i}} \) . The classes associated with \( x \) ... | No |
Proposition 32. Let \( \left( {{\mathrm{V}}_{i},{\rho }_{i}}\right) \) be the distinct (up to isomorphism) irreducible linear representations of \( \mathrm{G} \) over \( \mathrm{K} \), and let \( {\chi }_{i} \) be the corresponding characters. Then\n\n(a) The \( {\chi }_{i} \) form a basis of \( {\mathrm{R}}_{\mathrm{K... | It is clear that the \( {\chi }_{i} \) generate \( {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \) . On the other hand, if \( i \neq j \) we have \( {\operatorname{Hom}}^{G}\left( {{V}_{i},{V}_{j}}\right) = 0 \) . But in general, if \( V \) and \( W \) have characters \( {\chi }_{V} \) and \( {\chi }_{W} \) , we h... | Yes |
Proposition 33. In order that a linear representation of \( \mathrm{G} \) over \( \mathrm{C} \) be realizable over \( \mathrm{K} \), it is necessary and sufficient that its character belong to \( {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \) . | The condition is obviously necessary. Suppose conversely that it is satisfied, and let \( \chi \) be the character of the given representation. In view of prop. 32, we have \( \chi = \sum {n}_{i}{\chi }_{i} \), with \( {n}_{i} \in \mathbf{Z} \), and we obtain:\n\n\[ \left\langle {\chi ,{\chi }_{i}}\right\rangle = {n}_{... | Yes |
Lemma 12. We have \( d \cdot {\overline{\mathbf{R}}}_{\mathbf{K}}\left( \mathrm{G}\right) \subset {\mathbf{R}}_{\mathbf{K}}\left( \mathrm{G}\right) \) . | First, let \( \mathrm{V} \) be a linear representation of \( \mathrm{G} \) over \( \mathrm{L} \) with character \( \chi \) ; by restricting scalars we can consider \( \mathrm{V} \) as a \( \mathrm{K} \) -vector space (of dimension \( d \) times as large) and even as a linear representation of \( \mathrm{G} \) over \( \... | Yes |
Theorem 24 (Brauer). If \( \mathrm{K} \) contains the \( m \) th roots of unity, then \( {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \) \( = \mathrm{R}\left( \mathrm{G}\right) \) . | Let \( \chi \in \mathrm{R}\left( \mathrm{G}\right) \) . By th. 20 of 10.5, we can write \( \chi \) in the form\n\n\[ \chi = \sum {n}_{i}{\operatorname{Ind}}_{{\mathrm{H}}_{i}}^{\mathrm{G}}\left( {\varphi }_{i}\right) ,\;\left( {{n}_{i} \in \mathbf{Z}}\right) \]\n\nwhere the \( {\varphi }_{i} \) are characters of degree... | Yes |
Theorem 25. In order that a class function \( f \) on \( \mathbf{G} \), with values in \( \mathbf{L} \), belong to \( \mathrm{K}{ \otimes }_{\mathbf{Z}}\mathrm{R}\left( \mathrm{G}\right) \), it is necessary and sufficient that\n\n(*) \n\n\[ \n{\sigma }_{t}\left( {f\left( s\right) }\right) = f\left( {s}^{t}\right) \;\te... | Let \( \rho \) be a representation of \( \mathrm{G} \) with character \( \chi \) . For \( s \in \mathrm{G} \), the eigenvalues \( {\omega }_{i} \) of \( \rho \left( s\right) \) are \( m \) th roots of unity, and the eigenvalues of \( \rho \left( {s}^{t}\right) \) are the \( {\omega }_{i}^{t} \) . Thus we have \n\n\[ \n... | Yes |
In order that a class function \( f \) on \( \mathrm{G} \) with values in \( \mathrm{K} \) belong to \( \mathrm{K} \otimes {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \), it is necessary and sufficient that it be constant on the \( {\Gamma }_{\mathrm{K}} \) - classes of \( \mathrm{G} \) . | If \( f \in \mathrm{K} \otimes {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \), then \( f\left( s\right) \in \mathrm{K} \) for all \( s \in \mathrm{G} \), and formula \( \left( *\right) \) shows that \( f\left( s\right) = f\left( {s}^{t}\right) \) for all \( t \in {\Gamma }_{\mathbf{K}} \) . Hence \( f \) is const... | Yes |
Corollary 2. Let \( {\chi }_{i} \) be the characters of the distinct irreducible representations of \( \mathrm{G} \) over \( \mathrm{K} \) . Then the \( {\chi }_{i} \) form a basis for the space of functions on \( \mathrm{G} \) which are constant on \( {\Gamma }_{\mathrm{K}} \) -classes, and their number is equal to th... | This follows from cor. 1. | No |
Theorem 26. Let \( \mathrm{T} \) be the set of cyclic subgroups of \( \mathrm{G} \). Then the map\n\n\[ \mathbf{Q} \otimes \text{ Ind: }\underset{\mathrm{H} \in \mathrm{T}}{ \oplus }\mathbf{Q} \otimes {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{H}\right) \rightarrow \mathbf{Q} \otimes {\mathrm{R}}_{\mathrm{K}}\left( \mathr... | The two proofs given in Ch. 9 apply without change. The first is a duality argument; one must show that the mapping\n\n\[ \mathbf{Q} \otimes \text{Res:}\underset{H \in T}{ \oplus }\mathbf{Q} \otimes {\mathbf{R}}_{K}\left( G\right) \rightarrow \underset{H \in T}{ \oplus }\mathbf{Q} \otimes {\mathbf{R}}_{K}\left( H\right... | Yes |
Theorem 27. The map Ind: \( \underset{\mathrm{H} \in {\mathrm{X}}_{\mathrm{K}}}{ \oplus }{\mathrm{R}}_{\mathrm{K}}\left( \mathrm{H}\right) \rightarrow {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \) is surjective. | As in 10.5, we obtain theorem 27 from a more precise result, relative to a fixed prime number \( p \) | No |
Theorem 28. Let \( g = {p}^{n}l \) be the order of \( \mathrm{G} \), where \( \left( {p, l}\right) = 1 \) . The constant function \( l \) belongs to the image \( {\mathrm{V}}_{\mathrm{K}, p} \) of the map \[ \text{Ind:}\underset{\mathrm{H} \in {\mathrm{X}}_{\mathrm{K}}\left( p\right) }{ \oplus }{\mathrm{R}}_{\mathrm{K}... | The proof of this theorem is completely analogous to that of th. \( {18}^{\prime } \) (to which it reduces when \( \mathrm{K} \) is algebraically closed). We will give the proof in the next section and, for the time being, just indicate two consequences: | No |
Proposition 36. Let \( \varphi \) be a class function on \( \mathbf{G} \) . In order that \( \varphi \) belong to \( {R}_{\mathrm{K}}\left( \mathrm{G}\right) \), it is necessary and sufficient that, for each \( {\Gamma }_{\mathrm{K}} \) -elementary subgroup \( \mathrm{H} \) of \( \mathrm{G} \), we have \( {\operatornam... | Using th. 27, we have an identity\n\n\[ 1 = \mathop{\sum }\limits_{{\mathrm{H} \in {\mathrm{X}}_{\mathrm{K}}}}{\operatorname{Ind}}_{\mathrm{H}}^{\mathrm{G}}{f}_{\mathrm{H}},\;\text{ with }{f}_{\mathrm{H}} \in {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{H}\right) . \]\n\nMultiplying by \( \varphi \), this gives\n\n\[ \varph... | Yes |
Proposition 37. If each of the algebras \( \mathrm{K}\left\lbrack \mathrm{H}\right\rbrack ,\mathrm{H} \in {\mathrm{X}}_{\mathrm{K}} \), is quasisplit (cf. 12.2), the same is true of \( \mathrm{K}\left\lbrack \mathrm{G}\right\rbrack \) . | Let \( \varphi \in {\overline{\mathbf{R}}}_{\mathbf{K}}\left( G\right) \) . For \( H \in {X}_{\mathbf{K}} \), we have \( {\operatorname{Res}}_{H}^{G}\varphi \in {\overline{\mathbf{R}}}_{\mathbf{K}}\left( H\right) \), and \( {\overline{\mathbf{R}}}_{\mathbf{K}}\left( H\right) \) is equal to \( {R}_{K}\left( H\right) \) ... | Yes |
Lemma 14. There are finitely many prime ideals \( {\mathfrak{p}}_{1},\ldots ,{\mathfrak{p}}_{h} \) of A containing \( p \) . | The quotients \( A/{\mathfrak{p}}_{i} \) are finite fields of characteristic \( p \), and there exists an integer \( \mathrm{N} \) such that \( p\mathrm{\;A} \supset {\left( {\mathfrak{p}}_{1} \cap \cdots \cap {\mathfrak{p}}_{h}\right) }^{\mathrm{N}} \) . The \( {\mathfrak{p}}_{i} \) correspond to prime ideals of \( \m... | Yes |
Lemma 15. Let \( f \) be a function on \( \mathrm{G} \), constant on \( {\Gamma }_{\mathrm{K}} \) -classes, and with values in \( g \) A. Then \( f \) can be written in the form \[ f = \sum {\operatorname{Ind}}_{\mathrm{C}}^{\mathrm{G}}\left( {\varphi }_{\mathrm{C}}\right) ,\;\text{ with }{\varphi }_{\mathrm{C}} \in \m... | Let \( \varphi = f/g \) . In the notation of lemma 6, we have \[ f = \sum {\operatorname{Ind}}_{\mathrm{C}}^{\mathrm{G}}\left( {{\theta }_{\mathrm{C}} \cdot {\operatorname{Res}}_{\mathrm{C}}^{\mathrm{G}}\varphi }\right) \] and it remains only to show that \( {\varphi }_{\mathrm{C}} = {\theta }_{\mathrm{C}} \cdot {\oper... | Yes |
Lemma 16. Let \( x, y \in \mathrm{G} \) be elements whose \( {p}^{\prime } \) -components are \( {\Gamma }_{\mathrm{K}} \) -conjugate. If \( f \in \mathrm{A} \otimes {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \), then\n\n\[ f\left( x\right) \equiv f\left( y\right) \;\left( {\;\operatorname{mod.}\;{\mathfrak{p}}_... | We know that \( f \) is constant on \( {\Gamma }_{\mathbf{K}} \) -classes (cor. 1 of th. 25). So we can assume that \( x \) is the \( {p}^{\prime } \) -component of \( y \), in which case the same argument applies as in the proof of lemma 7. | Yes |
Lemma 17. Let \( x \) be a \( {p}^{\prime } \) -element of \( \mathrm{G} \), let \( \mathrm{C} \) be the cyclic subgroup generated by \( x \), let \( \mathrm{N}\left( x\right) \) be the set of \( y \in \mathrm{G} \) such that there exists \( t \in {\Gamma }_{\mathrm{K}} \) with \( {yx}{y}^{-1} = {x}^{t} \), and let \( ... | Assertion (a) is clear. To prove (b), it suffices to consider the case of an irreducible representation over \( \mathrm{K} \) . Such a representation can be obtained by choosing a homomorphism \( \chi : \mathrm{C} \rightarrow {\mathrm{L}}^{ * } \), taking as vector space the subfield \( {\mathrm{K}}_{\chi } \) generate... | Yes |
Lemma 18. Keep the notation of lemma 17. Then there exists \[ \psi \in \mathrm{A} \otimes {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{H}\right) \] such that the induced function \( {\psi }^{\prime } = {\operatorname{Ind}}_{\mathrm{H}}^{\mathrm{G}}\psi \) has the following properties: (i) \( {\psi }^{\prime }\left( x\right)... | Let \( c \) be the order of \( \mathrm{C} \), and let \( {\psi }_{\mathrm{C}} \) be the function defined on \( \mathrm{C} \) by \( {\psi }_{\mathrm{C}}\left( y\right) = c \) when \( y \) has the form \( {x}^{t} \), with \( t \in {\Gamma }_{\mathrm{K}} \), and \( {\psi }_{\mathrm{C}}\left( y\right) = 0 \) otherwise. The... | Yes |
Lemma 19. There exists \( \varphi \in \mathrm{A} \otimes {\mathrm{V}}_{\mathrm{K}, p} \) such that \( \varphi \left( x\right) \equiv 0\left( {\;\operatorname{mod.}\;{\mathfrak{p}}_{i}}\right) \) for each \( x \in \mathrm{G} \) and each \( i = 1,\ldots, h \) . | Let \( {\left( {x}_{\lambda }\right) }_{\lambda \in \Lambda } \) be a system of representatives for the \( {p}^{\prime } \) -regular \( {\Gamma }_{\mathrm{K}} \) -classes i.e., those consisting of \( {p}^{\prime } \) -elements. For each \( \lambda \in \Lambda \), the preceding lemma allows us to construct \( {\varphi }... | Yes |
Corollary 1. The number of isomorphism classes of irreducible representations of \( \mathbf{G} \) over \( \mathbf{Q} \) is equal to the number of conjugacy classes of cyclic subgroups of \( \mathbf{G} \) . | This follows from cor. 2 to th. 25. | No |
Corollary 2. The following properties are equivalent:\n\n(i) Each character of \( \mathrm{G} \) has values in \( \mathbf{Q} \) .\n\n(i') Each character of \( \mathbf{G} \) has values in \( \mathbf{Z} \) .\n\n(ii) Two elements of \( \mathrm{G} \) which generate the same subgroup are conjugate. | The equivalence of (i) and (i') comes from the fact that character values are algebraic integers, thus are elements of \( \mathbf{Z} \) whenever they belong to \( \mathbf{Q} \) . The equivalence of (i) and (ii) follows from th. 29. | No |
Theorem 30. Each element of \( {\mathrm{R}}_{\mathbf{O}}\left( \mathrm{G}\right) \) is a linear combination, with coefficients in \( \mathbf{Q} \), of characters \( {1}_{\mathrm{C}}^{\mathrm{G}} \), where \( \mathbf{C} \) runs over the set of cyclic subgroups of \( \mathbf{G} \) . | This amounts to saying that \( \mathbf{Q} \otimes {\mathbf{R}}_{\mathbf{O}}\left( \mathrm{G}\right) \) is generated by the \( {l}_{\mathrm{C}}^{\mathrm{G}} \) . Since \( \mathbf{Q} \otimes {\mathbf{R}}_{\mathbf{Q}}\left( G\right) \) is endowed with the nondegenerate bilinear form\n\n\[ \left( {\varphi ,\psi }\right) \m... | Yes |
Proposition 39. In order that \( \rho \) be of type 1,2, or 3, it is necessary and sufficient that the number\n\n\[ \n\left\langle {1,{\Psi }^{2}\left( \chi \right) }\right\rangle = \frac{1}{g}\mathop{\sum }\limits_{{s \in \mathrm{G}}}\chi \left( {s}^{2}\right) ,\;\text{ where }g = \operatorname{Card}\left( \mathrm{G}\... | Let \( {\chi }_{\sigma }^{2} \) (resp. \( {\chi }_{\lambda }^{2} \) ) be the character of the symmetric square (resp. the alternating square) of V. Then\n\n\[ \n{\chi }_{\sigma }^{2} = \frac{1}{2}\left( {{\chi }^{2} + {\Psi }^{2}\chi }\right) ,\;{\chi }_{\lambda }^{2} = \frac{1}{2}\left( {{\chi }^{2} - {\Psi }^{2}\chi ... | Yes |
Proposition 40. The family of all elements [E], with \( \mathrm{E} \in {\mathrm{S}}_{\mathrm{L}} \), is a basis for the group \( {\mathrm{R}}_{\mathrm{L}}\left( \mathrm{G}\right) \) . | Let \( R \) be the free \( Z \) -module with basis \( {S}_{L} \) . The family of the various [E], \( \mathrm{E} \in {\mathrm{S}}_{\mathrm{L}} \), defines a homomorphism \( \alpha : \mathrm{R} \rightarrow {\mathrm{R}}_{\mathrm{L}}\left( \mathrm{G}\right) \) . On the other hand, if \( \mathrm{F} \) is an \( \mathrm{L}\le... | Yes |
Lemma 20. Let \( \Lambda \) be a commutative ring, and \( \mathrm{P}{a\Lambda }\left\lbrack \mathrm{G}\right\rbrack \) -module. In order that \( \mathrm{P} \) be projective over \( \Lambda \left\lbrack \mathrm{G}\right\rbrack \), it is necessary and sufficient that it be projective over \( \Lambda \), and that there ex... | If \( \mathrm{P} \) is projective over \( \Lambda \left\lbrack \mathrm{G}\right\rbrack \) it is projective over \( \Lambda \) : this follows from the fact that \( \Lambda \left\lbrack \mathrm{G}\right\rbrack \) is \( \Lambda \) -free. Conversely, suppose that the underlying \( \Lambda \) -module \( {\mathrm{P}}_{0} \) ... | Yes |
Lemma 21. Suppose that \( \Lambda \) is a local ring, with residue field \( {k}_{\Lambda } = \Lambda /{\mathfrak{m}}_{\Lambda } \) . (a) Let \( \mathrm{P} \) be a \( \Lambda \) -free \( \Lambda \left\lbrack \mathrm{G}\right\rbrack \) module. In order that \( \mathrm{P} \) be \( \Lambda \left\lbrack \mathrm{G}\right\rbr... | If \( \mathrm{P} \) is \( \Lambda \left\lbrack \mathrm{G}\right\rbrack \) -projective, then \( \overline{\mathrm{P}} \) is \( {k}_{\Lambda }\left\lbrack \mathrm{G}\right\rbrack \) -projective. Conversely, if this condition is satisfied, the preceding lemma shows that there exists a \( {k}_{\Lambda } \) - endomorphism \... | Yes |
Every projective A[G]-module is a direct sum of projective indecomposable A[G]-modules; this decomposition is unique up to isomorphism. A projective indecomposable \( \mathrm{A}\left\lbrack \mathrm{G}\right\rbrack \) -module is characterized up to isomorphism by its reduction \( {\;\operatorname{mod}\;.}\mathfrak{m} \)... | This follows from the preceding proposition and known results for projective \( k\left\lbrack \mathrm{G}\right\rbrack \) -modules. | No |
Corollary 3. Reduction \( {\;\operatorname{mod}\;.}\mathfrak{m} \) defines an isomorphism from \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \) onto \( {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) ; this isomorphism maps \( {\mathrm{P}}_{\mathrm{A}}^{ + }\left( \mathrm{G}\right) \) onto \( {\mathrm{P}}_{k}^{ + }\... | As a result we may identify \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \) and \( {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) . | No |
Theorem 32. The image of \( {\overline{\mathrm{E}}}_{1} \) in \( {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) is independent of the choice of the stable lattice \( {\mathrm{E}}_{1} \) . | Let \( {\mathrm{E}}_{2} \) be a lattice of \( \mathrm{E} \) stable under \( \mathrm{G} \) . We must show that \( \left\lbrack {\overline{\mathrm{E}}}_{1}\right\rbrack = \left\lbrack {\overline{\mathrm{E}}}_{2}\right\rbrack \) in \( {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) . We begin with a special case:\n\nWe have \... | Yes |
Theorem 33. The homomorphism \( d : {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \rightarrow {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) is surjective. | The proof will be given in 17.3. | No |
Theorem 34. The homomorphism \( e : {\mathrm{P}}_{k}\left( \mathrm{G}\right) \rightarrow {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \) is a split injection. | When \( \mathrm{K} \) is sufficiently large, \( e \) is the transpose of \( d \) (cf. 15.4), and the fact that \( d \) is surjective implies that \( e \) is a split injection. In the general case, let \( {\mathrm{K}}^{\prime } \) be a finite sufficiently large extension of \( \mathrm{K} \), and let \( {k}^{\prime } \) ... | Yes |
Corollary 1. For each finite extension \( {\mathrm{K}}^{\prime } \) of \( \mathrm{K} \), the homomorphism\n\n\[ \n{\mathrm{P}}_{k}\left( \mathrm{G}\right) \overset{e}{ \rightarrow }{\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \rightarrow {\mathrm{R}}_{{\mathrm{K}}^{\prime }}\left( \mathrm{G}\right) \n\]\n\nis a sp... | The injectivity of \( e \) is equivalent to: | No |
Corollary 2. Let \( \mathrm{P} \) and \( {\mathrm{P}}^{\prime } \) be projective \( \mathrm{A}\left\lbrack \mathrm{G}\right\rbrack \) -modules. If the \( \mathrm{K}\left\lbrack \mathrm{G}\right\rbrack \) -modules \( \mathrm{K} \otimes \mathrm{P} \) and \( \mathrm{K} \otimes {\mathrm{P}}^{\prime } \) are isomorphic, the... | (Indeed we know that the equality \( \left\lbrack \mathrm{P}\right\rbrack = \left\lbrack {\mathrm{P}}^{\prime }\right\rbrack \) in \( {\mathrm{R}}_{\mathrm{A}}\left( \mathrm{G}\right) \simeq {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) is equivalent to \( \mathrm{P} \simeq {\mathrm{P}}^{\prime } \).) | Yes |
Corollary 1. The map \( c : {\mathrm{P}}_{k}\left( \mathrm{G}\right) \rightarrow {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) is injective, and its cokernel is a finite p-group. | The second assertion is immediate from th. 35 ; the first then follows, since \( {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) and \( {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) are free \( \mathbf{Z} \) -modules with the same rank, namely \( \operatorname{Card}\left( {\mathrm{S}}_{k}\right) \) . | Yes |
Corollary 2. If two projective \( k\left( \mathrm{G}\right) \) -modules have the same composition factors (with multiplication), they are isomorphic. | This is a restatement of the injectivity of \( c \) . | No |
Corollary 3. Assume \( \mathrm{K} \) is sufficiently large. The Cartan matrix \( \mathrm{C} \) is then symmetric, and the corresponding quadratic form is positive definite. The determinant of \( \mathrm{C} \) is a power of \( p \) . | The quadratic form in question is\n\n\[ x \mapsto \langle x, c\left( x\right) {\rangle }_{k} = \langle x, d\left( {e\left( x\right) }\right) {\rangle }_{k} = \langle e\left( x\right), e\left( x\right) {\rangle }_{\mathrm{K}},\;x \in {\mathrm{P}}_{k}\left( \mathrm{G}\right) .\n\]\n\nSince the form \( \langle a, b{\rangl... | Yes |
Theorem 37. Let \( {\mathrm{K}}^{\prime } \) be a finite extension of \( \mathrm{K} \) . In order that an element of \( {\mathrm{R}}_{{\mathrm{K}}^{\prime }}\left( \mathrm{G}\right) \) belong to the image of \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) = {\mathrm{P}}_{k}\left( \mathrm{G}\right) \) under \( e \)... | For the proof, see 17.5. | No |
Lemma 22. Let \( x \in {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \), and let \( n \geq 1 \) be an integer. If \( {nx} \in {\mathrm{P}}_{\mathrm{A}}^{ + }\left( \mathrm{G}\right) \), we have \( x \in {\mathrm{P}}_{\mathrm{A}}^{ + }\left( \mathrm{G}\right) \) . | This is clear: if \( r = \operatorname{Card}\left( {\mathrm{S}}_{k}\right) \), then \( {\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) \) can be identified with \( {\mathbf{Z}}^{r} \) and \( {\mathrm{P}}_{\mathrm{A}}^{ + }\left( \mathrm{G}\right) \) with \( {\mathrm{N}}^{r} \), cf. 14.3 and 14.4. | No |
Proposition 44. Let \( {\mathrm{K}}^{\prime } \) be a finite extension of \( \mathrm{K} \), and let \( {\mathrm{A}}^{\prime } \) be the ring of integers of \( {\mathrm{K}}^{\prime } \) . Assume the following two conditions on an element \( x \) of \( {\mathrm{R}}_{{\mathrm{K}}^{\prime }}\left( \mathrm{G}\right) \) :\n\... | Let \( \mathrm{N} = \left\lbrack {{\mathrm{K}}^{\prime } : \mathrm{K}}\right\rbrack = \left\lbrack {{\mathrm{A}}^{\prime } : \mathrm{A}}\right\rbrack \) . Let \( {\mathrm{E}}^{\prime } \) be a projective \( {\mathrm{A}}^{\prime }\left\lbrack \mathrm{G}\right\rbrack \) -module with image \( {nx} \) in \( {\mathrm{R}}_{{... | Yes |
Proposition 45. Suppose the following condition is satisfied:\n\n(R) There exists a finite extension \( {\mathrm{K}}^{\prime } \) of \( \mathrm{K} \), with residue field \( {k}^{\prime } \), such that \( d\left( {{\mathrm{R}}_{{\mathrm{K}}^{\prime }}^{ + }\left( \mathrm{G}\right) }\right) = {\mathrm{R}}_{{k}^{\prime }}... | By prop. 44 it is enough to prove that\n\n\[ e\left( {{\mathrm{P}}_{\mathrm{A}}^{ + }\left( \mathrm{G}\right) }\right) = e\left( {{\mathrm{P}}_{\mathrm{A}}\left( \mathrm{G}\right) }\right) \cap {\mathrm{R}}_{\mathrm{K}}^{ + }\left( \mathrm{G}\right) \]\n\nwhen \( \mathrm{K} \) is sufficiently large, in which case condi... | Yes |
Theorem 38. (Fong-Swan). Suppose that \( \\mathrm{G} \) is p-solvable, i.e., has a composition series whose factors are either p-groups or groups of order prime to p. Then \( \\mathrm{G} \) satisfies conditions \( \\left( \\mathrm{R}\\right) \) and \( \\left( {\\mathrm{R}}^{\\prime }\\right) \) above. | For the proof, see 17.6. | No |
Proposition 46. Let \( \mathrm{E} \) be a simple \( \mathrm{K}\left\lbrack \mathrm{G}\right\rbrack \) -module, and let \( \mathrm{P} \) be a lattice in \( \mathrm{E} \) stable under \( \mathrm{G} \) . Assume that the dimension \( \mathrm{N} \) of \( \mathrm{E} \) is divisible by the largest power \( {p}^{n} \) of \( p ... | First of all, since \( \mathrm{N} \) is divisible by \( {p}^{n} \), the quotient \( \mathrm{N}/g \) belongs to the ring A. This enables us to apply Fourier inversion (6.2., prop. 11) without introducing any \ | No |
Theorem 39. Let \( \mathrm{X} \) be the set of all \( {\Gamma }_{\mathrm{K}} \) -elementary subgroups of \( \mathrm{G} \) (cf. 12.6). The homomorphisms \[ \text{Ind:}\underset{\mathrm{H} \in \mathrm{X}}{ \oplus }{\mathrm{R}}_{k}\left( \mathrm{H}\right) \rightarrow {\mathrm{R}}_{k}\left( \mathrm{G}\right) \] and \[ \tex... | Let \( {l}_{K} \) (resp. \( {l}_{k} \) ) denote the identity element of the ring \( {R}_{K}\left( G\right) \) (resp. \( {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) ). We have \( d\left( {\mathrm{l}}_{\mathrm{K}}\right) = {\mathrm{l}}_{k} \) . By th. 27 we can write \( {\mathrm{l}}_{\mathrm{K}} \) in the form \[ {\mathr... | Yes |
Corollary 2. The kernel of the homomorphism \( d : {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \rightarrow {\mathrm{R}}_{k}\left( \mathrm{G}\right) \) consists of those elements \( x \) whose virtual character \( {\chi }_{x} \) is zero on \( {\mathrm{G}}_{\text{reg }} \) . | (Since \( d \) is surjective, this gives an explicit description of \( {\mathbf{R}}_{k}\left( \mathbf{G}\right) \) as a quotient of \( {\mathrm{R}}_{\mathrm{K}}\left( \mathrm{G}\right) \) .) | No |
Corollary 3. The number of classes of simple \( k\left\lbrack \mathrm{G}\right\rbrack \) -modules is equal to the number of p-regular conjugacy classes of \( \mathbf{G} \) . | Proof of theorem 42.\n\n(a) We prove first that \( {\phi }_{\mathrm{E}}\left( {\mathrm{E} \in {\mathrm{S}}_{k}}\right) \) are linearly independent over \( \mathrm{K} \) . Indeed, suppose that we had a relation \( \sum {a}_{\mathrm{E}}{\phi }_{\mathrm{E}} = 0 \), with \( {a}_{\mathrm{E}} \in \mathrm{K} \), not all zero.... | Yes |
Theorem 44. Let \( l \) be a prime number unequal to the residue characteristic of \( \mathbf{E} \) .\n\n(i) The representations of Artin and Swan are realizable over the field \( {\mathbf{Q}}_{l} \) of \( l \) -adic numbers.\n\n(ii) There exists a projective \( {\mathbf{Z}}_{l}\left\lbrack \mathrm{G}\right\rbrack \) -... | It is enough to prove (ii); assertion (i) then follows, since \( {a}_{\mathrm{G}} \) is obtained from \( s{w}_{\mathrm{G}} \) by adding to it \( {u}_{\mathrm{G}} \), which is realizable over any field.\n\nFor this, we apply prop. 44, taking \( p = l,\mathrm{\;K} = {\mathbf{Q}}_{l}, n = g = \operatorname{Card}\left( \ma... | Yes |
Consider the circle \( \mathrm{V}\left( {{X}^{2} + {Y}^{2} - 1}\right) \) . Let \( X = {X}_{1} + i{X}_{2} \) and \( Y = {Y}_{1} + i{Y}_{2} \) . Then \( {\left( {X}_{1} + i{X}_{2}\right) }^{2} + {\left( {Y}_{1} + i{Y}_{2}\right) }^{2} = 1 \) . | Expanding and equating real and imaginary parts gives\n\n\[ \n{X}_{1}{}^{2} - {X}_{2}{}^{2} + {Y}_{1}{}^{2} - {Y}_{2}{}^{2} = 1,\;{X}_{1}{X}_{2} + {Y}_{1}{Y}_{2} = 0.\n\] | Yes |
We cannot leave this section of examples without at least briefly mentioning curves with singularities; an example is given by the alpha curve \( \mathrm{V}\left( {{Y}^{2} - {X}^{2}\left( {X + 1}\right) }\right) \) (Figure 2). | Separating real and imaginary parts of \( {Y}^{2} - {X}^{2}\left( {X + 1}\right) = 0 \) and setting \( {X}_{2} = 0 \) gives us a curve sketched in Figure 21. The two branches again meet at one point at infinity, \( {P}_{\infty } \), and the other curves \( {X}_{2} = \) constant fit together as in Figure 22. Topological... | Yes |
Any complex line and any complex circle in \( {\mathbb{P}}^{2}\left( \mathbb{C}\right) \) must intersect. | One can actually see, in Figure 7, how any parallel translate of the complex \( Y \) -plane along the \( {X}_{1} \) -axis still intersects the circle \( \mathrm{V}\left( {{X}^{2} + {Y}^{2} - 1}\right) \) , either in the \( \left( {{X}_{1},{Y}_{1}}\right) \) -plane (as we usually see the intersection), or in the hyperbo... | No |
Lemma 1.3. Let \( {S}_{1} \) and \( {S}_{2} \) be any two projective subspaces of \( {\mathbb{P}}^{n}\left( k\right) \) . Then\n\n\[ \operatorname{cod}\left( {{S}_{1} \cap {S}_{2}}\right) \leq \operatorname{cod}{S}_{1} + \operatorname{cod}{S}_{2}. \] | Proof. Any subspace \( {k}^{r + 1} \) has codimension \( n - r \) in \( {k}^{n + 1} \) ; therefore the associated subspace \( {\mathbb{P}}^{r}\left( k\right) \) has the same codimension \( n - r \) in \( {\mathbb{P}}^{n}\left( k\right) \) . Then apply the corresponding vector space theorem. | No |
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