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Lemma 5.1 Let \( W \) be an \( R \) -module (not necessarily of finite dimension over \( \mathbb{k} \) ) linearly generated over \( k \) by a set \( \mathcal{S} \) of simple \( R \) -submodules. Then for every proper \( R \) -submodule \( U \subsetneq W \), there exists a complementary \( R \) -submodule \( V \subset W... | Proof Since \( U \neq W \) and \( W \) is spanned by submodules \( S \in \mathcal{S} \), there exists \( S ⊄ U \) in \( S \) . Then the sum \( U + S \) is a direct sum, because \( S \) is simple, and therefore \( S \cap U \subsetneq S \) is zero. Write \( {S}^{\prime } \) for the set of semisimple submodules \( M \subs... | Yes |
Lemma 5.2 Let \( W \) be an \( R \) -module such that every nonzero proper submodule of \( W \) contains a simple R-submodule. \( {}^{5} \) Then \( W \) is semisimple if and only if every nonzero proper \( R \) -submodule \( U \subsetneq W \) admits a complementary \( R \) -submodule \( V \subset W \) such that \( W = ... | Proof Let every nonzero proper submodule \( M \subset W \) have a complementary submodule. Write \( S \) for the set of all semisimple submodules \( S \subseteq W \) partially ordered by the relation \( {S}_{1} < {S}_{2} \), meaning that \( {S}_{2} = {S}_{1} \oplus S \) for some \( S \in \mathcal{S} \) . The poset \( \... | Yes |
Theorem 5.1 (Semisimplicity Criteria) Let \( W \) be an R-module such that every \( R \) -submodule of \( W \) contains a finite-dimensional \( R \) -submodule. Then the following properties of \( W \) are equivalent:\n\n1. \( W \) is semisimple.\n\n2. \( W \) is linearly generated over \( \mathbb{k} \) by simple \( R ... | Proof For a finite-dimensional \( R \) -module \( U \), every \( R \) -submodule \( S \subset U \) of minimal nonzero dimension has to be simple. Thus, the assumption of Lemma 5.2 holds. Therefore, \( \left( 3\right) \Rightarrow \left( 1\right) \) . Certainly, \( \left( 1\right) \Rightarrow \left( 2\right) \) . Implica... | Yes |
Lemma 5.3 (Schur's Lemma) Every nonzero homomorphism of simple \( R \) - modules \( \varphi : U \rightarrow W \) is an isomorphism. If the ground field \( \mathbb{k} \) is algebraically closed, then the \( R \) -linear endomorphisms of a simple \( R \) -module \( U \) are exhausted by the scalar operators \( \lambda \c... | Proof Since \( \ker \varphi \subset U \) is \( R \) -invariant, either \( \ker \varphi = U \) or \( \ker \varphi = 0 \) . In the first case, \( \varphi = 0 \) . In the second, \( \operatorname{im}\varphi \subset W \) is a nonzero \( R \) -submodule, and therefore im \( \varphi = W \) . Hence, \( \varphi \) is bijective... | Yes |
Corollary 5.1 Let \( U, W \) be irreducible \( R \) -modules over an algebraically closed field. Then\n\n\[ \dim {\operatorname{Hom}}_{R}\left( {U, W}\right) = \left\{ \begin{array}{ll} 0 & \text{ if }U ≄ W \\ 1 & \text{ if }U \simeq W \end{array}\right. \] | Proof If there is an \( R \) -linear isomorphism \( \psi : U \simeq W \), then for every \( \varphi \in \operatorname{Hom}\left( {U, W}\right) \), the equality \( {\psi }^{-1}\varphi = \lambda \cdot {\operatorname{Id}}_{U} \) holds for some \( \lambda \in \mathbb{k} \) by Schur’s lemma. Hence \( \varphi = {\lambda \psi... | Yes |
Corollary 5.2 A quotient module of a semisimple \( R \) -module is semisimple. | Proof Let \( \pi : W \rightarrow U \) be an \( R \) -linear surjection. Then for every simple \( R \) - submodule \( S \subset W \), its image \( \pi \left( S\right) \subset U \) is either zero or simple. Thus if \( W \) is spanned by simple submodules, then so is \( U \) . | Yes |
Proposition 5.1 Under the assumptions of Theorem 5.1 on p. 102, an R-module W is semisimple if and only if for every submodule \( U \subset W \), there exists an \( R \) -linear endomorphism \( {\pi }_{U} \in {\operatorname{End}}_{R}\left( W\right) \) such that \( {\pi }_{U}^{2} = {\pi }_{U} \) and \( \operatorname{im}... | Proof We have seen in Example 15.3 of Algebra I that every linear endomorphism \( \pi : V \rightarrow V \) satisfying the relation \( {\pi }^{2} = \pi \) projects \( V \) onto \( \operatorname{im}\pi \) along \( \ker \pi \), i.e., \( V = \ker \pi \oplus \operatorname{im}\pi \) and \( \pi \left( u\right) = u \) for all ... | No |
Corollary 5.3 A submodule of a semisimple R-module is semisimple. | Proof Let \( W \) be a semisimple \( R \) -module, and \( L \subsetneq W \) an \( R \) -submodule. Then for every \( R \) -submodule \( U \subset L \), there exists an \( R \) -linear projector \( W \rightarrow U \) . Its restriction to \( L \) gives the required projector \( L \rightarrow U \) . | No |
Theorem 5.2 (Double Centralizer Theorem) Let \( V \) be a finite-dimensional vector space over \( k \), let \( A \subset \operatorname{End}\left( V\right) \) be an associative \( k \) -subalgebra, and let \( B = {\operatorname{End}}_{A}\left( V\right) \) . If \( V \) is a semisimple \( A \) -module, then \( {\operatorn... | Proof The inclusion \( A \subset {\operatorname{End}}_{B}\left( V\right) \) follows from the definition of centralizer. To establish the opposite inclusion, we fix some basis \( {e}_{1},{e}_{2},\ldots ,{e}_{n} \) of \( V \) over \( \mathbb{k} \) and for every \( \varphi \in {\operatorname{End}}_{B}\left( V\right) \), i... | Yes |
Corollary 5.4 (Burnside's Theorem) Let \( V \) be a finite-dimensional vector space over an algebraically closed field \( k \), and \( R \subset {\operatorname{End}}_{k}\left( V\right) \) a set of operators. If \( V \) is simple as an \( R \) -module, then the associative envelope \( \operatorname{Ass}\left( R\right) \... | Proof By Schur’s lemma, \( {}^{7}{\operatorname{End}}_{\operatorname{Ass}\left( R\right) }\left( V\right) = \mathbb{k} \) . Therefore, \( {\operatorname{End}}_{\mathbb{k}}\left( V\right) = \operatorname{Ass}\left( R\right) \) by Theorem 5.2. | No |
Proposition 5.2 For every A-linear map \( \varphi : V \rightarrow W \), the image of the \( U \) -isotypic component \( {V}_{U} \subset V \) belongs to the \( U \) -isotypic component \( {W}_{U} \subset W \) . In particular, \( {V}_{U} = V \cap {W}_{U} \) for every \( A \) submodule \( V \subset W \) . | Proof Every vector of the form \( \sum {\psi }_{i}\left( {u}_{i}\right) ,{\psi }_{i} \in {\operatorname{Hom}}_{A}\left( {U, V}\right) ,{u}_{i} \in U \), is mapped to \( \sum \varphi {\psi }_{i}\left( v\right) \), where \( \varphi {\psi }_{i} \in {\operatorname{Hom}}_{A}\left( {\bar{U}, W}\right) ,{u}_{i} \in U \) . | Yes |
Proposition 5.3 Over an algebraically closed ground field \( k \), the contraction map (5.9) is injective and therefore establishes the canonical isomorphism\n\n\[ \n{c}_{UW} : {\operatorname{Hom}}_{A}\left( {U, W}\right) \otimes U \simeq {W}_{U} \n\] | Proof Since \( {W}_{U} \) is linearly spanned by simple submodules isomorphic to \( U \), it follows from Lemma 5.1 applied to the set \( S \) of these submodules that \( {W}_{U} \) splits into a direct sum\n\n\[ \n{W}_{U} = {V}_{1} \oplus {V}_{2} \oplus \cdots \oplus {V}_{s},\text{ where }{V}_{i} \simeq U\text{ for al... | Yes |
Proposition 5.4 (Isotypic Decomposition) Let \( W = {\bigoplus }_{i}{V}_{i} \), where all \( {V}_{i} \) are simple A-modules. Then the sum of all the \( {V}_{i} \) that are isomorphic to \( U \) coincides with the \( U \) -isotypic component \( {W}_{U} \subset W \) . In particular, this sum does not depend on the choic... | Proof Since \( {\operatorname{Hom}}_{A}\left( {U, W}\right) = {\bigoplus }_{i}{\operatorname{Hom}}_{A}\left( {U,{V}_{i}}\right) \) and \( {\operatorname{Hom}}_{A}\left( {U,{V}_{j}}\right) = 0 \) for all \( {V}_{j} ≄ U \), the image of the canonical contraction (5.9) is contained in the sum of the \( {V}_{i} \) that are... | Yes |
Corollary 5.5 For every pair of finite-dimensional semisimple A-modules \( V, W \) over an algebraically closed field \( k \), one has\n\n\[ \dim {\operatorname{Hom}}_{A}\left( {V, W}\right) = \mathop{\sum }\limits_{\left\lbrack U\right\rbrack }{m}_{U}\left( U\right) \cdot {m}_{U}\left( W\right) = \dim {\operatorname{H... | Proof Let \( V = \oplus {V}_{i}, W = \oplus {W}_{j} \), where all \( {V}_{i},{W}_{j} \) are simple. By Schur’s lemma, the space \( {\operatorname{Hom}}_{A}\left( {{V}_{i},{W}_{j}}\right) \) is zero for \( {V}_{i} ≄ {W}_{j} \) and has dimension 1 for \( {V}_{i} \simeq {W}_{j} \) . Therefore, the space \( {\operatorname{... | Yes |
Lemma 5.4 Let \( G \) be a finite group of order \( \left| G\right| = n \) . Assume that \( \operatorname{char}k \nmid n \), and that the polynomial \( {t}^{n} - 1 \) completely splits over \( \mathbb{k} \) into a product of \( n \) linear factors. Then in every (not necessarily finite-dimensional) \( G \) -module \( V... | Proof Since \( {g}^{\left| G\right| } = e \) for all \( g \in G \), every operator \( g \in G \) in a linear representation of \( G \) is annihilated by the polynomial \( f\left( t\right) = {t}^{n} - 1 \) . By our assumption, \( f \) is a product\n\n\( {}^{8} \) That is, it takes every \( \xi : V \rightarrow \mathbb{k}... | Yes |
Theorem 5.3 (Pontryagin Duality) For every finite abelian group \( G \) and \( g \in G \) , the evaluation map\n\n\[ \n{\operatorname{ev}}_{g} : {G}^{ \land } \rightarrow \mathbb{k},\;\chi \mapsto \chi \left( g\right) ,\n\]\n\nis a multiplicative character of the Pontryagin dual group \( {G}^{ \land } \) . The map\n\n\... | Proof The first statement holds because\n\n\[ \n{\operatorname{ev}}_{g}\left( {{\chi }_{1}{\chi }_{2}}\right) = {\chi }_{1}\left( g\right) \cdot {\chi }_{2}\left( g\right) = {\operatorname{ev}}_{g}\left( {\chi }_{1}\right) \cdot {\operatorname{ev}}_{g}\left( {\chi }_{2}\right) .\n\]\n\nSince \( {\operatorname{ev}}_{{g}... | Yes |
Theorem 5.4 Every linear representation \( V \) of a finite group \( G \) over a field \( k \) with char \( k \nmid \left| G\right| \) is completely reducible. | Proof By Proposition 5.1 on p. 104, it is enough to show that every G-submodule \( U \subset V \) admits a \( G \) -linear projector \( {\pi }_{U} : V \rightarrow U \) . Recall \( {}^{12} \) that \( G \) acts on \( {\operatorname{Hom}}_{\mathbb{k}}\left( {V, U}\right) \) as \( g : \varphi \mapsto {g\varphi }{g}^{-1} \)... | Yes |
Lemma 5.5 Let \( \varrho : \mathbb{k}\left\lbrack G\right\rbrack \rightarrow \operatorname{End}\left( V\right) \) be a linear representation. If \( {m}_{\lambda }\left( V\right) = 0 \), then \( \rho \left( {I}_{\lambda }\right) = 0 \) | Proof For every \( v \in V \), the assignment \( x{e}_{\lambda } \mapsto \varrho \left( {x{e}_{\lambda }}\right) v \) gives a well-defined \( G \) -linear map \( {I}_{\lambda } \rightarrow V \) . By Proposition 5.2, the image of this map is contained in the \( \lambda \) -isotypic component of \( V \), which is zero by... | Yes |
Corollary 5.6 The multiplicity \( {m}_{\lambda }\left( {\mathbb{k}\left\lbrack G\right\rbrack }\right) \neq 0 \) for every irreducible \( G \) -module \( \lambda \) , that is, \( \mathbb{k}\left\lbrack G\right\rbrack = { \oplus }_{\lambda \in \operatorname{Irr}\left( G\right) }{I}_{\lambda } \) . | Proof If there exists an irreducible \( G \) -module \( W \) that does not appear in (5.22), then \( {m}_{\lambda }\left( W\right) = 0 \) for all \( \lambda \) from (5.22). It follows from Lemma 5.5 that \( \mathbb{k}\left\lbrack G\right\rbrack \) acts by zero on \( W \), i.e., \( W = 0 \) . | Yes |
Proposition 5.5 Every linear representation \( \varrho : \mathbb{k}\left\lbrack G\right\rbrack \rightarrow \operatorname{End}\left( V\right) \) maps every irreducible idempotent \( {e}_{\lambda },\lambda \in \operatorname{Irr}\left( G\right) \), to the \( \lambda \) -isotypic projector \( {\pi }_{\lambda } : V \rightar... | Proof By Lemma 5.5, the left multiplication by \( {e}_{\lambda } \) annihilates all ideals \( {I}_{\varrho } \) with \( \varrho \neq \lambda \) . This forces \( {e}_{\lambda } \) to act by zero in every irreducible representation \( \varrho \neq \lambda \) . By Schur’s lemma, the action of \( {e}_{\lambda } \) in the i... | Yes |
Corollary 5.7 The irreducible idempotents \( {e}_{\lambda } \) belong to \( Z\left( {\mathbb{k}\left\lbrack G\right\rbrack }\right) \) and are linearly independent over \( \mathbb{k} \) . In particular, \( \left| {\operatorname{Irr}\left( G\right) }\right| \leq \left| {\mathrm{{Cl}}\left( G\right) }\right| \) . | Proof Applying Proposition 5.5 to the left regular representation shows that left multiplication by \( {e}_{\lambda } \) acts identically on \( {I}_{\lambda } \) and annihilates all \( {I}_{\varrho } \) with \( \varrho \neq \lambda \) . Thus,\n\n\[ \n{e}_{\lambda }\mathop{\sum }\limits_{\varrho }{x}_{\varrho }{e}_{\var... | Yes |
Theorem 5.5 (Maschke's Theorem) Let \( G \) be a finite group and \( k \) an algebraically closed field with \( \operatorname{char}k \mid G \mid \) . Then an isomorphism of \( k \) -algebras is given by the map\n\n\[ \mathfrak{r} : \mathbb{k}\left\lbrack G\right\rbrack \rightarrow {\bigoplus }_{\lambda \in \operatornam... | Proof Let us first show that \( \mathfrak{r} \) is injective. If \( h \in \mathbb{k}\left\lbrack G\right\rbrack \) acts by the zero operator in all irreducible representations, then \( h \) is zero in every finite-dimensional representation, because every such representation splits into a direct sum of irreducible repr... | Yes |
Under the assumptions of Theorem 5.5, \( {m}_{\lambda }\left( {\mathbb{k}\left\lbrack G\right\rbrack }\right) = \dim {U}_{\lambda } \) for every \( \lambda \in \operatorname{Irr}\left( G\right) \), and\n\n\[ \mathop{\sum }\limits_{{\lambda \in \operatorname{Irr}\left( G\right) }}{\dim }^{2}{U}_{\lambda } = \left| G\rig... | The equality \( {m}_{\lambda }\left( {\mathbb{k}\left\lbrack G\right\rbrack }\right) = \dim {U}_{\lambda } \) was established in the proof of (5.23). Comparing the dimensions of both sides in (5.23) gives (5.24):\n\n\[ \left| G\right| = \dim \mathbb{k}\left\lbrack G\right\rbrack = \mathop{\sum }\limits_{{\lambda \in \o... | Yes |
In accordance with Sect. 5.4.2, the abelian group \( G = \mathbb{Z}/\left( 3\right) \) has three irreducible representations of dimension 1 over \( \mathbb{k} = \mathbb{C} \) . The generator \( g = \left\lbrack 1\right\rbrack \) acts in these representations as multiplication by \( 1,\omega \), and \( {\omega }^{2} \) ... | This agrees with Corollary 5.8 and the isotypic decomposition\n\n\[\n\mathbb{C}\left\lbrack G\right\rbrack \simeq \frac{\mathbb{C}\left\lbrack g\right\rbrack }{\left( {g}^{3} - 1\right) } \simeq \frac{\mathbb{C}\left\lbrack g\right\rbrack }{\left( g - 1\right) } \oplus \frac{\mathbb{C}\left\lbrack g\right\rbrack }{\lef... | Yes |
The decomposition from Example 2.3 on p. 33,\n\n\[ \n{V}^{\otimes 3} = {\operatorname{Sym}}^{3}\left( V\right) \oplus {\operatorname{Alt}}^{3}\left( V\right) \oplus {W}_{\Delta },\n\]\n\n(5.26)\n\nis the isotypic decomposition with respect to the action of \( {S}_{3} \) . The three symmetry types appearing here are cal... | The \( {S}_{3} \) -linear projectors on the components are provided by the operators \( {\operatorname{sym}}_{3},{\mathrm{{alt}}}_{3} \), and \( {\pi }_{\Delta } \) from Example 5.5. Thus, a tensor \( t \in {V}^{\otimes 3} \) is of Lie type if and only if it is annihilated by averaging over the action of a 3-cycle: \( ... | Yes |
Lemma 5.6 The linear span of the operators \( {f}^{\otimes n}, f \in \mathrm{{GL}}\left( V\right) \), coincides with the centralizer \( {\operatorname{End}}_{{S}_{n}}\left( {V}^{\otimes n}\right) \) of the action of \( {S}_{n} \) on \( {V}^{\otimes n} \) . | Proof The chain of canonical isomorphisms\n\n\[ \mathrm{{End}}\left( {V}^{\otimes n}\right) \simeq {{V}^{\otimes n}}^{ * } \otimes {V}^{\otimes n} \simeq {{V}^{ * }}^{\otimes n} \otimes {V}^{\otimes n} \simeq {\left( {V}^{ * } \otimes V\right) }^{\otimes n} \simeq \mathrm{{End}}{\left( V\right) }^{\otimes n} \]\n\niden... | Yes |
Proposition 5.6 All the Schur representations\n\n\\[ \n{\\mathbb{S}}^{\\lambda }V = {\\operatorname{Hom}}_{{S}_{n}}\\left( {{U}_{\\lambda },{V}^{\\otimes n}}\\right) \n\\]\n\nof \\( \\mathrm{{GL}}\\left( V\\right) \\) are irreducible. | Proof The isomorphism \\( {\\$ }^{\\lambda }V \\otimes {U}_{\\lambda } \\rightarrow {W}_{\\lambda } \\) from (5.28) transfers the action of \\( {S}_{n} \\) on \\( {W}_{\\lambda } \\) to the action \\( g : \\varphi \\otimes u \\mapsto \\varphi \\otimes \\left( {gu}\\right) \\) . Every linear operator \\( F \\in \\operat... | Yes |
Prove that every two matrix elements from different irreducible representations are orthogonal with respect to the standard Hermitian structure on \( {\mathbb{C}}^{G} \) provided by the inner product\n\n\[\n\\left( {{f}_{1},{f}_{2}}\\right) = {\\left| G\\right| }^{-1}\\mathop{\\sum }\\limits_{{g \\in G}}{f}_{1}\\left( ... | Hint: for every \( \\mathbb{k} \)-linear map between irreducible representations \( \\varphi : {U}_{\\lambda } \\rightarrow {U}_{\\varrho } \), the average\n\n\[\n{\\left| G\\right| }^{-1}\\mathop{\\sum }\\limits_{{g \\in G}}{g\\varphi }{g}^{-1} = {\\left| G\\right| }^{-1}\\mathop{\\sum }\\limits_{{g \\in G}}{g\\varphi... | No |
Proposition 6.1 (Plancherel’s Formula) For all \( f, g \in \mathbb{k}\left\lbrack G\right\rbrack \) , \n\n\[ \n\left( {f, g}\right) = \mathop{\sum }\limits_{{\lambda \in \operatorname{Irr}\left( G\right) }}\dim \left( {U}_{\lambda }\right) \cdot \operatorname{tr}\left( {\lambda \left( {fg}\right) }\right) .\n\] | Proof The trace of left multiplication by \( {fg} \) in the algebra \( { \oplus }_{\lambda \in \operatorname{Irr}\left( G\right) }\operatorname{End}\left( {U}_{\lambda }\right) \) equals the sum of the traces of left multiplications by \( \lambda \left( {fg}\right) \) in the algebras \( \operatorname{End}\left( {U}_{\l... | Yes |
For every \( \lambda \in \operatorname{Irr}\left( G\right) \), the linear expansion of \( {e}_{\lambda } \) through the group elements is\n\n\[ \n{e}_{\lambda } = \frac{\dim {U}_{\lambda }}{\left| G\right| }\mathop{\sum }\limits_{{g \in G}}\operatorname{tr}\left( {\lambda \left( {g}^{-1}\right) }\right) \cdot g.\n\] | Proof By formula (6.5), \( {e}_{\lambda } = {\left| G\right| }^{-1}\mathop{\sum }\limits_{{\mu \in \operatorname{Irr}\left( G\right) }}\left( {{g}^{-1},{e}_{\lambda }}\right) \cdot g \) . By the Plancherel formula, \( {}^{3} \)\n\n\[ \n\left( {{g}^{-1},{e}_{\lambda }}\right) = \mathop{\sum }\limits_{{\mu \in \operatorn... | Yes |
Let a group \( G \) act on \( {\mathbb{k}}^{n} \) by permutations of the standard basis vectors. Then the character of this action takes an element \( g \in G \) to the number of fixed points of the permutation provided by \( g \) . In particular, the values of the character of the left regular representation are | \[ {\chi }_{L}\left( g\right) = \left\{ \begin{array}{ll} \left| G\right| & \text{ for }g = e, \\ 0 & \text{ for }g \neq e. \end{array}\right. \] | Yes |
Example 6.2 (Irreducible Characters of \( {S}_{4} \) ) If a representation admits an explicit geometric description, its character usually can be computed by straightforward summation of the eigenvalues of rotations and reflections representing the group elements. For example, the five irreducible representations of \(... | The fourth row of this table was computed as follows. The trace of the identity equals the dimension of the representation. Since a lone transposition and a pair of disjoint transpositions act as rotations by \( {180}^{ \circ } \) about some lines, their eigenvalues are \( 1, - 1, - 1 \), and the trace equals -1 . A 3-... | Yes |
Lemma 6.1 For every linear representations \( V, W \) of a finite group \( G \) with characters \( {\chi }_{U},{\chi }_{V} \), one has\n\n\[{\chi }_{V \oplus W}\left( g\right) = {\chi }_{V}\left( g\right) + {\chi }_{W}\left( g\right)\]\n\n(6.15)\n\n\[{\chi }_{V \otimes W}\left( g\right) = {\chi }_{V}\left( g\right) {\c... | Proof Since every operator \( g \) in a finite group of linear operators is diagonalizable over an algebraically closed field of characteristic zero, there exist bases\n\n\[{v}_{1},{v}_{2},\ldots ,{v}_{n} \in V\;\text{ and }\;{w}_{1},{w}_{2},\ldots ,{w}_{m} \in W\]\n\nconsisting of eigenvectors of \( g \) . Write \( {\... | Yes |
Corollary 6.1 The character of an arbitrary linear representation \( V \) is a linear combination of the irreducible characters \( {\chi }_{\lambda },\lambda \in \operatorname{Irr}\left( G\right) \) with nonnegative integer coefficients: | \[ {\chi }_{V} = \mathop{\sum }\limits_{{\lambda \in \operatorname{Irr}\left( G\right) }}{m}_{\lambda }\left( V\right) \cdot {\chi }_{\lambda } \] where \( {m}_{\lambda }\left( V\right) = \dim {V}_{\lambda }/\dim {U}_{\lambda } \) is the multiplicity \( {}^{6} \) of the simple \( G \) -module \( {U}_{\lambda } \) in \(... | Yes |
Corollary 6.3 \( \dim {\operatorname{Hom}}_{G}\left( {V, W}\right) = \left( {{\chi }_{V},{\chi }_{W}}\right) \) for every pair of finite-dimensional G-modules \( V, W \) . | Proof Both sides are equal to \( \mathop{\sum }\limits_{{\lambda \in \operatorname{Irr}\left( G\right) }}{m}_{\lambda }\left( V\right) {m}_{\lambda }\left( W\right) \), where \( {m}_{\lambda }\left( M\right) \) means the multiplicity of the irreducible representation \( \lambda \) in a given \( G \) -module \( M \) . F... | Yes |
Corollary 6.4 The multiplicity of a simple \( G \) -module \( {U}_{\lambda } \) in an arbitrary \( G \) -module \( V \) can be computed by the formula \( {m}_{\lambda }\left( V\right) = \left( {{\chi }_{\lambda },{\chi }_{V}}\right) \) . | Proof Take the inner product of the character \( {\chi }_{\lambda } \) with both sides of formula (6.19) on p. 136, and use the orthonormality of irreducible characters. | No |
Corollary 6.5 A linear representation \( V \) is irreducible if and only if \( \left( {{\chi }_{V},{\chi }_{V}}\right) = 1 \) . | Proof It follows from Corollary 6.1 and the orthonormality of irreducible characters that\n\n\[ \left( {{\chi }_{V},{\chi }_{V}}\right) = \mathop{\sum }\limits_{{\lambda \in \operatorname{Irr}\left( G\right) }}{m}_{\lambda }^{2}\left( V\right) \]\n\nwhere all the multiplicities \( {m}_{\lambda }\left( V\right) \) are n... | Yes |
Proposition 6.4 (Transitivity of Induction) For every tower of subgroups \( K \subset H \subset G \) and every linear representation \( \varrho : K \rightarrow \mathrm{{GL}}\left( U\right) \), there is the canonical isomorphism of \( G \) -modules \( {\operatorname{ind}}_{H}^{G}{\operatorname{ind}}_{K}^{H}U \simeq {\op... | Proof Since for every \( G \) -module \( W \) there are the canonical isomorphisms\n\n\[ \n{\operatorname{Hom}}_{K}\left( {U, W}\right) \simeq {\operatorname{Hom}}_{H}\left( {{\operatorname{ind}}_{K}^{H}U, W}\right) \simeq {\operatorname{Hom}}_{G}\left( {{\operatorname{ind}}_{H}^{G}{\operatorname{ind}}_{K}^{H}U, W}\rig... | No |
Example 6.4 Let \( G = {S}_{3} \), and let \( H \subset {S}_{3} \) be the subgroup of order 2 generated by the transposition \( \sigma = |{12}\rangle \) . Then the elements of \( G/H \) can be represented by \( e,\tau ,{\tau }^{2} \), where \( \tau = |{123}\rangle \) is a 3-cycle. The representation \( W = \) ind \( \m... | \[ \sigma = \left( \begin{array}{lll} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right) \;\text{ and }\;\tau = \left( \begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right) \] in this basis. Therefore, \( W \) is isomorphic to the tautological \( {S}_{3} \) -module, which is the direct sum of tr... | Yes |
Proposition 6.5 If a group \( G \) has an abelian subgroup \( H \subset G \), then every simple G-module has dimension at most \( {}^{13}\left\lbrack {G : H}\right\rbrack \) . | Proof Let \( U \) be an irreducible representation of \( G \), and \( L \subset \operatorname{res}U \) an \( H \) -submodule of dimension 1. By Frobenius reciprocity, \( U \) has positive multiplicity in ind \( L \) . Hence, \( \dim U \leq \operatorname{dimind}L = \left\lbrack {G : H}\right\rbrack \) . | Yes |
Proposition 6.6 Assume that the intersection of a conjugacy class \( C \subset G \) with a subgroup \( H \subset G \) splits into \( m \) distinct classes with respect to conjugation by the elements of \( H \) :\n\n\[ C \cap H = {D}_{1} \sqcup {D}_{2} \sqcup \cdots \sqcup {D}_{m}. \]\n\nThen for every representation \(... | Proof For every \( g \in C \), the summands \( {g}_{v}V \) in the decomposition (6.28) are permuted under the action of \( g \), and a nonzero contribution to the value \( {\chi }_{\text{ind }V}\left( g\right) \) is made only by those summands \( {g}_{v}V \) that are mapped to itself by \( g \) . The inclusion \( g\lef... | Yes |
Problem 6.12 (The Heisenberg Group Over \( {\mathbb{F}}_{2} \) ) Write \( H \) for the group generated by \( {4n} + 4 \) elements \( \pm 1, \pm {u}_{1},\ldots , \pm {u}_{{2n} + 1} \) constrained by the relations \[ {u}_{i}^{2} = - 1,\;{u}_{i}{u}_{j} = - {u}_{j}{u}_{i} \] | Verify that \( H \) consists of \( {2}^{{2n} + 2} \) distinct elements \( \pm {u}_{I} = \pm {u}_{{i}_{1}}{u}_{{i}_{2}}\cdots {u}_{{i}_{k}} \), where \( I = \left\{ {{i}_{1},{i}_{2},\ldots ,{i}_{k}}\right\} \) runs through the increasing subsets in \( \{ 1,2,\ldots ,\left( {n + 1}\right) \} \) and \( {u}_{\varnothing } ... | No |
Lemma 7.1 Let \( U, T \) be standard fillings of shapes \( \mu ,\lambda \) with the same weight \( \left| \lambda \right| = \left| \mu \right| \) . If \( \mu \) does not strictly dominate \( \lambda \), then either there are two numbers in the same row of \( T \) and in the same column of \( U \), or \( \lambda = \mu \... | Proof Suppose that the elements of every row in \( T \) are in different columns of \( U \) . Since the elements from the top row of \( T \) are distributed among different columns of \( U \), the inequality \( {\lambda }_{1} \leq {\mu }_{1} \) holds, and there exists \( {q}_{1} \in {C}_{U} \) moving all the elements f... | Yes |
Corollary 7.1 A permutation \( g \in {S}_{n} \) is factorized as \( g = {pq} \) with \( p \in {R}_{T}, q \in {C}_{T} \) if and only if the elements of every row in \( T \) appear in different columns of \( {gT} \) . Such a factorization is unique if it exists. | Proof If \( U = {pqT} \), where \( p \in {R}_{T}, q \in {C}_{T} \), then all elements of every row in \( T \) are in different columns of \( U \), because \( q \) shifts these elements along the columns and \( p \) permutes the resulting set of shifted elements within itself. Conversely, if every row of \( T \) is dist... | Yes |
Lemma 7.2 The vector space\n\n\[ \n{E}_{T}\overset{\text{ def }}{ = }\left\{ {\sigma \in \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \mid \forall p \in {R}_{T},\forall q \in {C}_{T},{p\sigma q} = \operatorname{sgn}\left( q\right) \cdot \sigma }\right\} \]\n\nhas dimension 1 and is spanned by the Young symmetrizer \( {s... | Proof Let us show that every element \( \sigma = \mathop{\sum }\limits_{{g \in {S}_{n}}}{x}_{g}g \in {E}_{T} \) is equal to \( {x}_{e} \cdot {s}_{T} \) . The equality \( {p\sigma q} = \operatorname{sgn}\left( q\right) \cdot \sigma \) means that \( {x}_{pgq} = \operatorname{sgn}\left( q\right) \cdot {x}_{g} \) for all \... | Yes |
For every filling \( T \), the equalities \( {s}_{T} \cdot \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {s}_{T} = \mathbb{C} \cdot {s}_{T} \) and \( {s}_{T}^{2} = {n}_{\lambda } \cdot {s}_{T} \) hold, where\n\n\[ \n{n}_{\lambda } = \frac{n!}{\dim \left( {\mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {s}_{T}}\r... | It follows from (7.4) and (7.5) that for every \( x \in \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \), the element \( {s}_{T} \cdot x \cdot {s}_{T} \) possesses the property (7.5) and therefore lies in the dimension-one subspace\n\n\( {E}_{T} = \mathbb{C} \cdot {s}_{T} \) from Lemma 7.2. In particular, \( {s}_{T}^{2} ... | Yes |
Lemma 7.4 If the shape of a filling \( T \) is lexicographically bigger than the shape of a filling \( U \), then\n\n\[ \n{r}_{T} \cdot \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {c}_{U} = {c}_{U} \cdot \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {r}_{T} = {s}_{T} \cdot \mathbb{C}\left\lbrack {S}_{n}\right... | Proof It is enough to check that \( {r}_{T} \cdot g \cdot {c}_{U} = {c}_{U} \cdot g \cdot {r}_{T} = 0 \) for all \( g \in {S}_{n} \) . To begin with, let \( g = e \) . Then by Lemma 7.1, there are two elements lying in the same row of \( T \) and column of \( U \) . The transposition \( \tau \) of these elements belong... | Yes |
Theorem 7.1 For every standard filling \( T \), the representation of \( {S}_{n} \) by left multiplication in the left ideal\n\n\[ \n{V}_{T} \cong \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {s}_{T} \subset \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \n\]\n\nis irreducible. Two such representations \( {V}_{T},{V}... | Proof Let \( W \subset {V}_{T} \) be an \( {S}_{n} \) -submodule. Write \( \pi : \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \rightarrow W \) for an \( {S}_{n} \) -linear projection, and let \( w = \pi \left( 1\right) \in W \) . Then \( \pi \left( x\right) = \pi \left( {x \cdot 1}\right) = x \cdot \pi \left( 1\right) =... | Yes |
Lemma 7.5 The representations of \( {S}_{n} \) by left multiplication in the ideals \( {V}_{T} = \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {s}_{T} \) and \( {V}_{T}^{\prime } = \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {s}_{T}^{\prime } \) are isomorphic. | Proof Right multiplication by \( {c}_{T} \) and \( {r}_{T} \) assigns homomorphisms of the left \( {S}_{n} \) -modules\n\n\[ \n{V}_{T}^{\prime } = \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {c}_{T}{r}_{T}\xrightarrow[{x \cdot {r}_{T} \leftrightarrow x}]{x \mapsto x \cdot {c}_{T}}\mathbb{C}\left\lbrack {S}_{n}\ri... | Yes |
Theorem 7.2 The classes of the irreducible representations \( {V}_{\lambda } \) and \( {V}_{{\lambda }^{\prime }} \) corresponding to the transposed Young diagrams \( \lambda \) and \( {\lambda }^{t} \) are obtained from each other by taking the tensor product with the sign representation. | Proof Let us fix some standard filling \( T \) of shape \( \lambda \) and the transposed filling \( {T}^{t} \) of shape \( {\lambda }^{t} \) . Then \( {R}_{{T}^{t}} = {C}_{T},{C}_{{T}^{t}} = {R}_{T} \), and\n\n\[ \n{s}_{{T}^{t}} = \mathop{\sum }\limits_{{p \in {R}_{T}}}\mathop{\sum }\limits_{{q \in {C}_{T}}}\operatorna... | Yes |
Proposition 7.1 Let \( {m}_{\lambda } = {m}_{{\lambda }_{1}}{m}_{{\lambda }_{2}}\cdots {m}_{{\lambda }_{n}} \) be the standard monomial basis \( {}^{3} \) of the \( \mathbb{Z} \) -module of symmetric polynomials in \( {x}_{1},{x}_{2},\ldots ,{x}_{n} \), and \[ {p}_{\eta }\left( x\right) = {p}_{{\eta }_{1}}{p}_{{\eta }_... | Proof The \( {n}_{i} \) th power of the \( i \) th Newton polynomial is expanded as \[ {p}_{i}{\left( x\right) }^{{n}_{i}} = {\left( {x}_{1}^{i} + {x}_{2}^{i} + \cdots + {x}_{n}^{i}\right) }^{{n}_{i}} = \mathop{\sum }\limits_{{\mathop{\sum }\limits_{j}{\varrho }_{ij} = {n}_{i}}}\frac{{n}_{i}!}{{\varrho }_{i1}!{\varrho ... | Yes |
Lemma 7.6 If the shape \( \lambda \) of a filling \( T \) does not strictly dominate a Young diagram \( \mu \), then\n\n\[ \n{c}_{T}{M}_{\mu } = \left\{ \begin{array}{ll} 0 & \text{ for }\mu \neq \lambda , \\ \mathbb{C} \cdot {v}_{T} & \text{ for }\mu = \lambda . \end{array}\right.\n\] | Proof Let \( U \) be a standard filling of shape \( \mu \) . If there is a transposition \( \tau \in {R}_{U} \cap {C}_{T} \) , then\n\n\[ \n{c}_{T}\{ U\} = {c}_{T}\{ {\tau U}\} = {c}_{T} \cdot \tau \{ U\} = - {c}_{T}\{ U\} .\n\]\n\n(7.12)\n\nHence, \( {c}_{T}\{ U\} = 0 \) . If there are no transpositions in \( {R}_{U} ... | Yes |
Theorem 7.3 The Specht module \( {S}_{\lambda } \) is simple and belongs to the class \( {V}_{\lambda } \), i.e., is isomorphic to the left ideal \( \mathbb{C}\left\lbrack {S}_{n}\right\rbrack \cdot {s}_{T} \), where \( T \) is a standard filling of shape \( \lambda \) . | Proof Let \( T \) be a standard filling of shape \( \lambda \) . Assume that \( {S}_{\lambda } = V \oplus W \) is a direct sum of \( {S}_{n} \) -modules. Since \( {c}_{T}{S}_{\lambda } \subset {c}_{T} \cdot {M}_{\lambda } = \mathbb{C} \cdot {v}_{T} \) by Lemma 7.6, and \( {c}_{T} \) maps each of the submodules \( V, W ... | Yes |
Corollary 7.2 The multiplicity of the simple submodule \( {S}_{\mu } \) in the tabloid module \( {M}_{\lambda } \) may be nonzero only if \( \mu \trianglerighteq \lambda \) . For all \( \lambda \), the multiplicity of \( {S}_{\lambda } \) in \( {M}_{\lambda } \) equals 1 . | Proof Since \( {c}_{T} \) sends the whole of \( {M}_{\lambda } \) inside \( {S}_{\lambda } \subset {M}_{\lambda } \) and acts nontrivially on \( {S}_{\lambda } \) , there is exactly one simple submodule isomorphic to \( {S}_{\lambda } \) in \( {M}_{\lambda } \) . If there exists an \( {S}_{n} \) -linear injection \( {S... | Yes |
Theorem 7.4 The vectors \( {v}_{T} \), where \( T \) runs through the standard tableaux of shape \( \lambda \), form a basis of the Specht module \( {S}_{\lambda } \) . In particular, \( \dim {S}_{\lambda } = {d}_{\lambda } \) equals the number of standard Young tableaux \( {}^{6} \) of shape \( \lambda \) . | Proof Let us first check that the \( {d}_{\lambda } \) vectors \( {v}_{T} \) are linearly independent. The linear expression of the vector \( {v}_{T} = \mathop{\sum }\limits_{{q \in {C}_{T}}}\operatorname{sgn}\left( q\right) \cdot \{ {qT}\} \) through the basis vectors \( \{ U\} \) of \( {M}_{\lambda } \) has the form\... | Yes |
Lemma 7.7 The graded commutative ring \( \Re \) is the ring of polynomials with integer coefficients in the countable set of variables \( \left\lbrack {\mathbb{1}}_{k}\right\rbrack, k \in \mathbb{N} \), the classes of trivial \( {S}_{k} \) -modules of dimension one. The isomorphism classes of tabloid representations\n\... | Proof It follows from Corollary 7.2 that the transition matrix from the isomorphism classes \( \left\lbrack {M}_{\lambda }\right\rbrack \) to the classes of irreducible representations \( \left\lbrack {S}_{\lambda }\right\rbrack \) is integer upper unitriangular. Therefore, the classes \( \left\lbrack {M}_{\lambda }\ri... | Yes |
Corollary 7.5 (Ramification Rules) Let \( {S}_{n} \subset {S}_{n + 1} \) be embedded as the stabilizer of some element. Then the representation of \( {S}_{n + 1} \) induced by an irreducible representation \( {S}_{\lambda } \) of \( {S}_{n} \) is a direct sum of simple modules \( {S}_{\mu } \), each taken with multipli... | Proof Since \( \left\lbrack {\operatorname{ind}\left( {S}_{\lambda }\right) }\right\rbrack = \left\lbrack {S}_{\lambda }\right\rbrack \cdot \left\lbrack {\mathbb{1}}_{1}\right\rbrack \), the first statement follows from the Littlewood-Richardson rule and Pieri’s formula, \( {}^{18} \) which expands \( {s}_{\lambda } \c... | Yes |
Corollary 7.6 (Frobenius Formula for Characters of \( {S}_{n} \) ) The value of an irreducible character \( {\chi }_{\lambda } \) of a symmetric group \( {S}_{n} \) on a conjugacy class \( {C}_{\mu } \subset {S}_{n} \) equals each of the following three coinciding integers:\n\n- the coefficient of \( {z}_{\mu }^{-1} \c... | Proof The first item follows directly from Theorem 7.5. To prove the second, recall that the Newton polynomials \( {p}_{\mu } \) form an orthogonal basis of \( \mathbb{Q} \otimes \Lambda \) with \( \left\langle {{p}_{\mu },{p}_{\mu }}\right\rangle = {z}_{\mu } \) . Therefore, the coefficient of \( {z}_{\mu }^{-1} \cdot... | Yes |
Exercise 7.9 (Hook Length Formula) Given a Young diagram \( \lambda \) and a cell \( a \in \lambda \) , the hook of \( a \) is the \( \Gamma \) -shaped subdiagram \( \Gamma \left( a\right) \subset \lambda \) formed by the cell \( a \) and all the cells below \( a \) in the column of \( a \) and to the right of \( a \) ... | For example, the hook lengths of the cells in the Young diagram \( \lambda = \left( {4,2,1}\right) \) are \n\nand therefore, the Specht representation \( {S}_{\left( 4,2,1\right) } \) of the symmetric group \( {S}_{7... | No |
Problem 7.13 Prove that the multiplicity of \( {S}_{\lambda } \) in \( {S}_{\mu } \otimes {S}_{v} \) is equal to\n\n\[ \mathop{\sum }\limits_{\eta }{z}_{\eta }^{-1}{\chi }_{\lambda }\left( {C}_{\eta }\right) {\chi }_{\mu }\left( {C}_{\eta }\right) {\chi }_{v}\left( {C}_{\eta }\right) \] | Verify that it becomes \( {\delta }_{\mu, v} \) for \( \lambda = \left( n\right) \), one row of length \( n \), and \( {\delta }_{\mu ,{v}^{t}} \) for \( \lambda = \left( {1}^{n}\right) \) , one column of height \( n \), where the Kronecker symbol \( {\delta }_{\alpha ,\beta } \) equals 1 for \( \alpha = \beta \) and 0... | No |
Proposition 8.1 All the standard \( {\mathfrak{{sl}}}_{2} \) -modules \( {V}_{n} \) are simple. | Proof Write an arbitrary vector \( v \in {V}_{n} \) as a linear combination of basis vectors \( {e}_{k} = {x}^{k}{y}^{n - k} \), and let \( m \) be the maximal index such that the coefficient of \( {e}_{m} \) in the expansion of \( v \) is not zero. It follows from formula (8.7) that \( {X}^{k}{Y}^{m}v \) is a nonzero ... | Yes |
Lemma 8.1 Let \( W \) be an \( {\mathfrak{{sl}}}_{2} \) -module, and let \( {W}_{\lambda }\overset{\text{ def }}{ = }\{ w \in W \mid {Hw} = {\lambda w}\} ,\lambda \in \mathbb{k} \), be an eigensubspace (possibly zero) of \( H \) . Then \( X\left( {W}_{\lambda }\right) \subset {W}_{\lambda + 2} \) and \( Y\left( {W}_{\l... | Proof If \( {Hw} = {\lambda w} \), then it follows from the commutation relations \( {HX} - {XH} = {2X} \) and \( {HY} - {YH} = - {2Y} \) that \( {HXw} = {XHw} + {2Xw} = \left( {\lambda + 2}\right) {Xw} \) and\n\n\[ \n{HYw} = {YHw} - {2Yw} = \left( {\lambda - 2}\right) {Yw}.\n\] | Yes |
Lemma 8.2 Every finite-dimensional \( {\mathfrak{{sl}}}_{2} \) -module over an algebraically closed field k possesses a primitive vector. | Proof Since \( \mathbb{k} \) is algebraically closed, we have \( \operatorname{Spec}H \neq \varnothing \), and there exists a weight vector \( v \neq 0 \) . The nonzero vectors in the chain \( v,{Xv},{X}^{2}v,\ldots \) are the eigenvectors of \( H \) with strictly increasing eigenvalues. Since they are linearly indepen... | Yes |
Lemma 8.3 Let \( W \) be a finite-dimensional \( {\mathfrak{{sl}}}_{2} \) -module over a field of characteristic zero. Then every primitive vector in \( W \) has nonnegative integer weight, and the \( {\mathfrak{{sl}}}_{2} \) -orbit of every primitive vector of weight \( m \) is isomorphic to the standard \( {\mathfrak... | Proof Let \( {Hv} = {\lambda v} \) and \( {Xv} = 0 \) for a nonzero vector \( v \in W \) . By Lemma 8.1, nonzero vectors of the chain \( v,{Yv},{Y}^{2}v,\ldots \) are the eigenvectors of \( H \) with eigenvalues \( \lambda ,\left( {\lambda - 2}\right) ,\left( {\lambda - 4}\right) ,\ldots \) Hence, there exists \( m \in... | Yes |
Theorem 8.1 The simple finite-dimensional \( {\mathfrak{{sl}}}_{2} \) -modules over a field \( k \) of characteristic zero are exhausted (up to isomorphism) by the standard modules \( {V}_{n} \) from Example 8.2. | Proof Let \( \overline{\mathbb{k}} \subset \mathbb{k} \) be the algebraic closure \( {}^{2} \) of the field \( \mathbb{k} \) . The tensor product of vector spaces \( \bar{V} = \overline{\mathbb{k}} \otimes V \) over \( \mathbb{k} \) is a vector over \( \overline{\mathbb{k}} \) with the action of \( \overline{\mathbb{k}... | "No" |
Example 8.3 (Isomorphism \( {V}_{n}^{ * } \simeq {V}_{n} \) ) Let \( {V}_{n}^{ * } \) be the dual \( {\mathfrak{{sl}}}_{2} \) -module to the standard irreducible \( \mathfrak{s}{\mathfrak{l}}_{2} \) -module \( {V}_{n} \), and suppose that the vectors \( {e}_{k}^{ * } \in {V}_{n}^{ * } \) form the dual basis to the stan... | \[ X\left( {e}_{k}^{ * }\right) = - \left( {n - k + 1}\right) {e}_{k - 1}^{ * },\;Y\left( {e}_{k}^{ * }\right) = - \left( {k + 1}\right) {e}_{k + 1}^{ * },\;H\left( {e}_{k}^{ * }\right) = - \left( {{2k} - n}\right) {e}_{k}^{ * }.\]\n\nHence, the \( {\mathfrak{{sl}}}_{2} \) -module \( {V}_{n}^{ * } \) has the same weigh... | Yes |
Lemma 8.4 Let \( V \) be an \( {\mathfrak{{sl}}}_{2} \) -module and \( U \subset V \) an \( {\mathfrak{{sl}}}_{2} \) -submodule of codimension 1. Then there exists a trivial \( \mathfrak{{sl}} \) 2-submodule \( L \simeq {V}_{0} \) in \( V \) such that \( V = U \oplus L \) . | Proof Note that every 1-dimensional \( {\mathfrak{{sl}}}_{2} \) -module \( L \) is trivial, because the algebra \( {\operatorname{End}}_{\mathbb{k}}\left( L\right) \simeq \mathbb{k} \) is commutative and therefore \( H = \left\lbrack {X, Y}\right\rbrack = 0,{2X} = \left\lbrack {H, X}\right\rbrack = 0 \) , \( {2Y} = \le... | No |
Theorem 8.2 Every finite-dimensional \( \mathfrak{{sl}} \) 2-module \( V \) is semisimple, i.e., splits into a direct sum of standard simple \( {\mathfrak{{sl}}}_{2} \) -modules \( {V}_{m} \) from Example 8.2 on p. 176. | Proof Let \( U \subset V \) be a proper nonzero \( {\mathfrak{{sl}}}_{2} \) -submodule. It is enough to show that there exists an \( {\mathfrak{{sl}}}_{2} \) -linear projector \( \pi : V \rightarrow U \) . The vector spaces\n\n\[ W = \\left\\{ {\\varphi : V \\rightarrow U{\\left| \\varphi \\right| }_{U} = \\lambda {\\o... | Yes |
Example 8.4 (Exterior Squares of Standard Simple Modules) Since the standard basis vector \( {e}_{k} = {x}^{k}{y}^{n - k} \) in \( {V}_{n} \) is an eigenvector of \( H \) with eigenvalue \( {2k} - n \) for all \( 0 \leq k \leq n \), the products \( {e}_{ij}\overset{\text{ def }}{ = }{e}_{i} \land {e}_{j},0 \leq i < j \... | We conclude that the \( {\mathfrak{{sl}}}_{2} \) -isotypic decomposition of \( {\Lambda }^{2}{V}_{n} \) is\n\n\[ \n{\Lambda }^{2}{V}_{n} \simeq {V}_{{2n} - 2} \oplus {V}_{{2n} - 6} \oplus {V}_{{2n} - {10}} \oplus \cdots = {\bigoplus }_{s = 0}^{\left\lbrack \left( n - 1\right) /2\right\rbrack }{V}_{2\left( {n - {2s} - 1... | Yes |
Problem 8.5 Show that the \( \mathfrak{s}{\mathfrak{l}}_{2} \) -invariant isomorphism \( {V}_{1} ⤳ {V}_{1}^{ * } \) is provided by the right correlation map \( {}^{9} \) of the skew-symmetric bilinear form det on \( {V}_{1} \), which sends a degree-one polynomial \( \beta \left( {x, y}\right) = {b}_{1}x + {b}_{2}y \in ... | \[ \det \left( {*,\beta }\right) : {V}_{1} \rightarrow \mathbb{k},\;\alpha \left( {x, y}\right) = {a}_{1}x + {a}_{2}y \mapsto \det \left( {\alpha ,\beta }\right) = \det \left( \begin{array}{ll} {a}_{1} & {b}_{1} \\ {a}_{2} & {b}_{2} \end{array}\right) . \] | Yes |
Every poset \( {}^{3}M \) can be considered a small category whose objects are the elements \( m \in M \) and whose arrows are the inequalities in \( M \) | \[ {\operatorname{Hom}}_{M}\left( {n, m}\right) = \left\{ \begin{array}{l} \text{ one element for }n \leq m, \\ \varnothing \text{ otherwise. } \end{array}\right. \] The composition of arrows \( k \leq \ell \) and \( \ell \leq n \) is the arrow \( k \leq n \) . The associativity and existence of the identity endomorphi... | Yes |
Every associative algebra \( A \) with unit \( e \in A \) over a commutative ring \( K \) can be viewed as a small category with just one object \( e \) and the set of morphisms \( \operatorname{Hom}\left( {e, e}\right) = A \), where the composition is the multiplication in \( A \) . | Conversely, associated with every small category \( \mathcal{C} \) and commutative ring \( K \) is the associative \( K \) -algebra of arrows \( {}^{5}K\left\lbrack \mathcal{C}\right\rbrack \) , the free \( K \) -module with basis Mor \( C \) and the \( K \) -bilinear multiplication defined on the basis vectors by the ... | No |
Example 9.5 (Geometric Realization of Combinatorial Simplices) The geometric realization functor from the simplicial category to the category of topological spaces \( \Delta \rightarrow \mathcal{T}{op},\left\lbrack n\right\rbrack \mapsto {\Delta }^{n} \), sends every combinatorial \( n \) -simplex to the regular \( n \... | \[ {\Delta }^{n} = \left\{ {\left( {{x}_{0},{x}_{1},\ldots ,{x}_{n}}\right) \in {\mathbb{R}}^{n + 1}\mid \sum {x}_{v} = 1,{x}_{v} \geq 0}\right\} ,\] the convex hull of heads of the standard basis vectors \( {e}_{0},{e}_{1},\ldots ,{e}_{n} \) in \( {\mathbb{R}}^{n + 1} \) . Under the geometric realization, an order-pre... | Yes |
Example 9.7 (Simplicial Sets) A presheaf of sets \( X : {\Delta }^{\text{opp }} \rightarrow \) Set on the whole simplicial category is called a simplicial set. Every simplicial set \( X \) also possesses the geometric realization \( \left| X\right| \) glued from regular simplices \( {\Delta }_{x}^{n}, x \in {X}_{n} \),... | topological direct product \( {}^{13}\mathop{\prod }\limits_{{n \geq 0}}{X}_{n} \times {\Delta }^{n} \) by the equivalence relation generated by all the identifications\n\n\[ \left( {x,{\varphi }_{ * }s}\right) \sim \left( {{\varphi }^{ * }x, s}\right) ,\;\varphi : \left\lbrack n\right\rbrack \rightarrow \left\lbrack m... | Yes |
Write vec \( = {ve}{c}_{\mathrm{k}} \) for the category of all finite-dimensional vector spaces over a field \( \mathbb{k} \) and \( \mathbb{k} \) -linear maps between them. Write \( \mathcal{C} \subset \) vec for the small full subcategory formed by coordinate vector spaces \( {\mathbb{k}}^{n} \) for all \( n \in {\ma... | sends a vector space \( V \) to the coordinate space \( {\mathbb{k}}^{\dim V} \) and a linear map \( \varphi : V \rightarrow W \) to the composition \n\[ \nF\left( \varphi \right) = {f}_{W} \circ \varphi \circ {f}_{V}^{-1} : {\mathbb{k}}^{\dim V} \rightarrow {\mathbb{k}}^{\dim W}, \n\] \nwhich can be treated as the mat... | Yes |
Proposition 9.1 A functor \( G : \mathcal{C} \rightarrow \mathcal{D} \) is an equivalence of categories if and only if \( G \) is fully faithful \( {}^{25} \) and essentially surjective, meaning that for every \( Y \in \mathrm{{Ob}}D \) , there exist \( X \in \mathrm{{Ob}}\mathcal{C} \), depending on \( Y \), and an is... | Proof We will prove the \ | No |
Lemma 9.1 (Yoneda Lemma for Presheaves) Let \( F : {\mathcal{C}}^{\text{opp }} \rightarrow \) Set be a presheaf of sets on a category \( \mathcal{C} \) . There is a bijection\n\n\[ F\left( A\right) \simeq {\operatorname{Hom}}_{\operatorname{PreSh}\left( \mathcal{C}\right) }\left( {{h}_{A}, F}\right) \]\n\n(9.11)\n\nfun... | Proof For every natural transformation (9.12), object \( X \in \mathrm{{Ob}}\mathcal{C} \), and arrow\n\n\[ \varphi : X \rightarrow A, \]\n\none has the commutative diagram (9.8), \n\n(9.13)\n\nwhose upper map takes ... | Yes |
Corollary 9.1 The prescriptions \( X \mapsto {h}_{X} \) and \( X \mapsto {h}^{X} \) assign fully faithful functors \( \mathcal{C} \hookrightarrow \operatorname{PreSh}\left( \mathcal{C}\right) \) and \( {\mathcal{C}}^{\text{opp }} \hookrightarrow \) Fun \( \left( {\mathcal{C},\text{Set}}\right) \) respectively. In parti... | Proof Apply the Yoneda lemmas to the functors \( F = {h}_{B} \) and \( F = {h}^{B} \) . | No |
Corollary 9.2 If a functor \( {F}^{\prime } : \mathcal{C} \rightarrow \) Set (respectively a presheaf \( F : {\mathcal{C}}^{\mathrm{{opp}}} \rightarrow \) Set) is corepresented (respectively represented), then its corepresenting (respectively representing) object \( A \in \mathrm{{Ob}}\mathcal{C} \) is unique up to can... | Proof Given natural isomorphisms \( \beta {\alpha }^{-1} : {h}^{A} \simeq {h}^{B} \) and \( {\beta }^{-1}\alpha : {h}^{B} \simeq {h}^{A} \), then by Corollary 9.1, there exist unique isomorphisms \( \psi : B \simeq A,\varphi : A \simeq B \) such that \( \beta {\alpha }^{-1} = {\psi }^{ * } \) and \( {\beta }^{-1}\alpha... | Yes |
Example 9.13 (Direct Product \( A \times B \) ) The direct product \( A \times B \) of objects \( A, B \in \mathrm{{Ob}}\mathcal{C} \) in an arbitrary category \( \mathcal{C} \) is defined as the representing object for the presheaf \( {\mathcal{C}}^{\text{opp }} \rightarrow \) Set, \( Y \mapsto \operatorname{Hom}\left... | \[ {\beta }_{Y} : \operatorname{Hom}\left( {Y, A \times B}\right) ⤳ \operatorname{Hom}\left( {Y, A}\right) \times \operatorname{Hom}\left( {Y, B}\right) \] functorial in \( Y \in \mathrm{{Ob}}\mathcal{C} \) . For \( Y = A \times B \), it produces the pair of arrows \[ A < \xrightarrow[]{{\pi }_{A}}A \times B\xrightarro... | Yes |
Exercise 9.20 Establish the isomorphism\n\n\[ \n{\operatorname{Hom}}_{R - \mathcal{M}{od}}\left( {R \otimes E, M}\right) \simeq {\operatorname{Hom}}_{\text{Set }}\left( {E, G\left( M\right) }\right) \n\]\n\n(9.16)\n\nfunctorial in \( M \in \mathrm{{Ob}}R \) - \( \mathcal{M} \) od, \( E \in \mathrm{{Ob}} \) Set. | The isomorphism (9.16) means that the functor \( \operatorname{Set} \rightarrow R \) - \( \mathcal{M}{od}, E \mapsto R \otimes E \), is the left adjoint to the forgetful functor \( G : R \) - \( \mathcal{M}{od} \rightarrow \) Set. The natural transformation\n\n\[ \n{\varrho }_{E} : E \hookrightarrow G\left( {A \otimes ... | Yes |
Proposition 9.2 For every two rings \( R, S \) and \( S \) - \( R \) -bimodule \( M \), the functor\n\n\[ R - \mathcal{M}{od} \rightarrow S - \mathcal{M}{od},\;X \mapsto M{ \otimes }_{R}X, \]\n\nis left adjoint to the functor\n\n\[ {h}^{M} : S\text{-}\mathcal{M}\text{od} \rightarrow R\text{-}\mathcal{M}\text{od},\;Y \m... | Proof The map from the left- to the right-hand side of (9.20) is constructed as follows. An \( S \) -linear homomorphism \( \varphi : M{ \otimes }_{R}X \rightarrow Y \) produces the family of maps\n\n\[ {\psi }_{x} : M \rightarrow Y,\;m \mapsto \varphi \left( {m \otimes x}\right) \]\n\ndepending on \( x \in X \) . Ever... | Yes |
Example 9.16 (Induced and Coinduced Modules) Let \( B \) be an arbitrary ring with unit and \( A \subset B \) a subring with the same unit. Every left \( B \) -module \( X \) can be viewed as a left \( A \) -module. This leads to the restriction functor \( {}^{32} \)\n\n\[ \text{res :}B - \mathcal{M}\text{od} \rightarr... | The left \( B \) -module coind \( Y\overset{\text{ def }}{ = }{\operatorname{Hom}}_{A}\left( {B, Y}\right) \) is called coinduced by the left \( A \) -module \( Y \) . Thus, the coinduction functor coind : \( A \) - \( \mathcal{M}{od} \rightarrow B \) - \( \mathcal{M}{od}, Y \mapsto \operatorname{coind}Y \), is right a... | No |
Proposition 9.3 A functor \( G : D \rightarrow \mathcal{C} \) admits the left adjoint functor \( F : \mathcal{C} \rightarrow D \) if and only if for every \( X \in \mathrm{{Ob}}\mathcal{C} \), the functor\n\n\[ \n{h}_{G}^{X} : \mathcal{D} \rightarrow \text{ Set },\;Y \mapsto {\operatorname{Hom}}_{\mathcal{C}}\left( {X,... | Proof The \ | No |
If a category \( C \) possesses a terminal object, direct products of all sets of objects, and equalizers of all pairs of arrows with common source and target, then \( C \) is complete. | We will prove the first statement; the second follows from it by reversing the arrows. Given a diagram \( X : \mathcal{N} \rightarrow \mathcal{C} \), we have to find the universal set of arrows \( {\varphi }_{v} \) with a common source and targets at \( {X}_{v} \) satisfying the equations \( {\varphi }_{\mu } = {x}_{\m... | Yes |
Corollary 9.3 The categories Set, Top, \( \mathcal{A}b,{\mathcal{{Mod}}}_{K}, R - \mathcal{{Mod}},\mathcal{{Mod}} - R,\mathcal{{Grp}},\mathcal{C}{mr} \) are bicomplete, meaning that they are both complete and cocomplete. | Proof This follows from Exercises 9.28-9.30. | No |
Example 9.23 (Open Neighborhoods and Stalks of Presheaves) In the category \( {\mathcal{U}}^{ * }\left( X\right) \) of open subsets of a topological space \( X \), every family of open sets closed with respect to intersections forms a cofiltered diagram. For example, all open sets containing a given subset \( Z \subset... | By the above construction, it is formed by the equivalence classes of pairs \( {s}_{U} \), where \( U \supset Z \) is an open neighborhood of \( Z \) and \( {s}_{U} \in F\left( U\right) \) is a section of \( F \) over \( U \) modulo the relation \( {s}_{U} \sim {s}_{W} \), meaning that \( {\left. {s}_{U}\right| }_{V} =... | Yes |
Proposition 9.7 If a functor \( F : \mathcal{C} \rightarrow \mathcal{D} \) is left adjoint to a functor \( G : \mathcal{D} \rightarrow \mathcal{C} \) , then \( F \) commutes with limits and \( G \) commutes with colimits. | Proof Since F2G, we have the following chain of isomorphisms functorial in \( D \in \operatorname{Ob}D \) :\n\n\[ \n{\mathrm{{Hom}}}_{D}\left( {F\left( {\mathrm{{colim}}\;X}\right), D}\right) \simeq {\mathrm{{Hom}}}_{\mathcal{C}}\left( {\;\mathrm{{colim}}\;X, G\left( D\right) }\right) \simeq {\mathrm{{Hom}}}_{\mathit{{... | No |
Corollary 9.5 Let \( X, Y : \mathcal{N} \rightarrow \mathcal{A}b \) be two diagrams of abelian groups, and\n\n\[ f : X \rightarrow Y \]\n\na natural transformation provided by the homomorphisms\n\n\[ {f}_{v} : {X}_{v} \rightarrow {Y}_{v},\;v \in \mathrm{{Ob}}\mathcal{N}. \]\n\nWrite \( K = \ker f \) and \( C = \operato... | Proof Since a (co)kernel is the (co)limit of a diagram, \( {}^{50} \) it commutes with (co) limits.\n\n\( {}^{50} \) Namely, the (co) kernel of a homomorphism is the (co)equalizer of this homomorphism and the zero homomorphism. | No |
Corollary 9.6 Let \( N \) be a right module over an arbitrary ring \( S \) . Then the functor \( S \) - \( \mathcal{M} \) od \( \rightarrow \mathcal{A}b, X \mapsto N{ \otimes }_{S}X \) commutes with the colimits of the diagrams of left S-modules. In particular,\n\n\[ \operatorname{coker}\left( {{\operatorname{Id}}_{N}{... | Proof Proposition 9.2 on p. 207 applied to the rings \( S \) and \( R = \mathbb{Z} \) shows that the functor\n\n\[ S\text{-}\mathcal{M}{od} \rightarrow \mathcal{A}b,\;X \mapsto N{ \otimes }_{S}X\text{,}\]\n\nis left adjoint to the functor \( \mathcal{A}b \rightarrow S \) - \( \mathcal{M}{od}, Y \mapsto {\operatorname{H... | Yes |
Problem 9.16 Prove that the abelian groups \( \mathbb{Q} \) and \( \mathbb{Q}/\mathbb{Z} \) are injective \( \mathbb{Z} \) -modules. | For every \( A \in \mathrm{{Ob}}\mathcal{A}b \) and \( a \in A \), prove that there exists a homomorphism of abelian groups \( \psi : A \rightarrow \mathbb{Q}/\mathbb{Z} \) such that \( \psi \left( a\right) \neq 0 \) . | No |
Lemma 10.1 (Characterization of Integral Elements) The following properties of an element \( b \in B \) in a ring extension \( A \subset B \) are equivalent:\n\n(1) \( {b}^{m} = {a}_{1}{b}^{m - 1} + \cdots + {a}_{m - 1}b + {a}_{m} \) for some \( m \in \mathbb{N} \) and \( {a}_{1},{a}_{2},\ldots ,{a}_{m} \in A \) .\n\n(... | Proof The implications \( \left( 1\right) \Rightarrow \left( 2\right) \Rightarrow \left( 3\right) \) are obvious. We will show that (3) implies (1). Fix some \( {e}_{1},{e}_{2},\ldots ,{e}_{m} \) spanning \( M \) over \( A \) and write \( Y \in {\operatorname{Mat}}_{m}\left( A\right) \) for the matrix of the \( A \) -l... | Yes |
Example 10.1 ( \( \mathbb{Z} \) is Integrally Closed in \( \mathbb{Q} \) ) Let \( A = \mathbb{Z}, B = \mathbb{Q} \) . If a fraction \( p/q \in \mathbb{Q} \) with coprime \( p, q \in \mathbb{Z} \) satisfies a monic polynomial equation\n\n\[ \frac{{p}^{m}}{{q}^{m}} = {a}_{1}\frac{{p}^{m - 1}}{{q}^{m - 1}} + \cdots + {a}_... | \[ {p}^{m} = {a}_{1}q{p}^{m - 1} + \cdots + {a}_{m - 1}{q}^{m - 1}p + {a}_{m}{q}^{m} \] is divisible by \( q \) . Since \( p, q \) are coprime, we conclude that \( q = \pm 1 \) . Hence, \( \mathbb{Z} \) is integrally closed in \( \mathbb{Q} \) . | Yes |
Let a finite group \( G \) act on a ring \( B \) by ring automorphisms, and let \( {B}^{G}\overset{\text{ def }}{ = }\{ a \in B \mid \forall g \in {Gga} = a\} \) be the subring of \( G \) -invariants. Then \( B \) is integral over \( {B}^{G} \) . | Indeed, write \( {b}_{1},{b}_{2},\ldots ,{b}_{n} \) for the \( G \) -orbit of an arbitrary element \( b = {b}_{1} \in B \) . Then \( b \) is a root of the monic polynomial\n\n\[ f\left( t\right) = \prod \left( {t - {b}_{i}}\right) \in {B}^{G}\left\lbrack t\right\rbrack \]\n\nas required in the first property of Lemma 1... | Yes |
Proposition 10.1 Let \( A \subset B \) be an extension of rings, and \( {\bar{A}}_{B} \subset B \) the integral closure of \( A \) in \( B \) . Then \( {\bar{A}}_{B} \) is a subring of \( B \), and for every ring extension \( B \subset C \) , every element \( c \in C \) integral over \( {\bar{A}}_{B} \) is integral ove... | Proof If elements \( p, q \in B \) satisfy the monic polynomial equations\n\n\[ \n{p}^{m} = {x}_{1}{p}^{m - 1} + \cdots + {x}_{m - 1}p + {x}_{m}, \n\]\n\n\[ \n{q}^{n} = {y}_{1}{q}^{n - 1} + \cdots + {y}_{n - 1}q + {y}_{n}, \n\]\n\nfor some \( {x}_{v},{y}_{\mu } \in A \), then the products \( {p}^{i}{q}^{j} \), where\n\... | Yes |
Proposition 10.2 (Gauss-Kronecker-Dedekind lemma) Let \( A \subset B \) be an extension of rings, and \( f, g \in B\left\lbrack x\right\rbrack \) monic polynomials of positive degree. Then all coefficients of the product \( {fg} \) are integral over \( A \) if and only if all coefficients of the polynomials \( f, g \) ... | Proof Let \( C \supset B \) be an extension of rings such that the polynomials \( f, g \) are completely factorizable in \( C\left\lbrack x\right\rbrack \) as \( f\left( x\right) = \prod \left( {x - {\alpha }_{v}}\right) \) and \( g\left( x\right) = \prod \left( {x - {\beta }_{\mu }}\right) \) for some \( {\alpha }_{v}... | Yes |
Proposition 10.3 Let \( A \subset B \) be an integral extension of rings. If \( B \) is a field, then \( A \) is a field too. Conversely, if \( A \) is a field and \( B \) has no zero divisors, then \( B \) is a field. | Proof Let \( B \) be an integral field over \( A \) . Then for every nonzero \( a \in A \), the inverse element \( {a}^{-1} \in B \) satisfies a monic polynomial equation\n\n\[ \n{a}^{-m} = {\alpha }_{1}{a}^{1 - m} + \cdots + {\alpha }_{m - 1}{a}^{-1} + {\alpha }_{0}\n\]\n\nfor some \( {\alpha }_{v} \in A \) . Multipli... | Yes |
Example 10.3 (Quadratic Algebraic Integers) Every field \( K \supset \mathbb{Q} \) of degree 2 has a form \( K = \mathbb{Q}\left\lbrack \sqrt{d}\right\rbrack = \mathbb{Q}\left\lbrack x\right\rbrack /\left( {{x}^{2} - d}\right) \), where \( d \in \mathbb{Z} \) is square-free and differs from 0,1. | Indeed, let \( \zeta \in K \smallsetminus \mathbb{Q} \) . Then 1, \( \zeta \) form a basis of \( K \) over \( \mathbb{Q} \), and \( {\zeta }^{2} = {b\zeta } + c \) for some \( b, c \in \mathbb{Q} \) . Hence, \( \zeta = a + b\sqrt{d} \) for some \( a, b \in \mathbb{Q} \) and \( d \in \mathbb{Z} \) such that \( b \neq 0 ... | Yes |
Corollary 10.2 Under the conditions of Corollary 10.1, let \( B \supset {Q}_{A} \) be a ring extending \( {Q}_{A} \) . If an element \( b \in B \) is integral over \( A \), then the minimal polynomial \( {}^{6} \) of b over \( {Q}_{A} \) lies in \( A\left\lbrack x\right\rbrack \) . | Proof Since \( b \) is integral over \( A \), there exists a monic polynomial \( f \in A\left\lbrack x\right\rbrack \) such that \( f\left( b\right) = 0 \) . Then the minimal polynomial of \( b \) over \( {Q}_{A} \) divides \( f \) in \( {Q}_{A}\left\lbrack x\right\rbrack \), and the quotient is also monic. It remains ... | Yes |
Theorem 10.2 Let \( G \) be a finite group, \( A \vartriangleleft G \) an abelian normal subgroup, and \( \varrho : \mathbb{C}\left\lbrack G\right\rbrack \rightarrow \) End \( W \) a complex irreducible representation. Then \( \dim W \) divides the index \( \left\lbrack {G : A}\right\rbrack \) . | Proof Consider the isotypic decomposition of the restriction of \( \varrho \) on \( A \) ,\n\n\[ \n\operatorname{res}W = {\bigoplus }_{\chi \in {A}^{ \land }}{W}_{\chi }\n\]\n\nwhere \( {W}_{\chi } \) is a direct sum of 1-dimensional representations in which the abelian group \( A \) acts by means of the same multiplic... | Yes |
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