Chapter 18

The group \(S_{3}\). Let \(S_{3}\) be the symmetric group on 3 elements. (a) Show that there are three conjugacy classes. (b) There are two characters of dimension 1, on \(S_{3} / A_{3}\). (c) Let \(d_{i}(i=1,2,3)\) be the dimensions of the irreducible characters. Since \(\sum d_{i}^{2}=6\), the third irreducible character has dimension 2. Show that the third representation can be realized by considering a cubic equation \(X^{3}+a X+b=0\), whose Galois group is \(S_{3}\) over a field \(k .\) Let \(V\) be the \(k\) vector space generated by the roots. Show that this space is 2 -dimensional and gives the desired representation, which remains irreducible after tensoring with \(k^{2}\). (d) Let \(G=S_{3}\). Write down an idempotent for each one of the simple components of \(\mathbf{C}[G] .\) What is the multiplicity of each irreducible representation of \(G\) in the regular representation on \(\mathrm{C}[G] ?\)

The groups \(S_{4}\) and \(A_{4}\). Let \(S_{4}\) be the symmetric group on 4 elements. (a) Show that there are 5 conjugacy classes. (b) Show that \(A_{4}\) has a unique subgroup of order 4, which is not cyclic, and which is normal in \(S_{4}\). Show that the factor group is isomorphic to \(S_{3}\), so the representations of Exercise 1 give rise to representations of \(S_{4}\). (c) Using the relation \(\sum d_{i}^{2}=\\#\left(S_{4}\right)=24\), conclude that there are only two other irreducible characters of \(S_{4}\), each of dimension 3 . (d) Let \(X^{4}+a_{2} X^{2}+a_{1} X+a_{0}\) be an irreducible polynomial over a field \(k\), with Galois group \(S_{4}\). Show that the roots generate a 3 -dimensional vector space \(V\) over \(k\), and that the representation of \(S_{4}\) on this space is irreducible, so we obtain one of the two missing representations. (e) Let \(\rho\) be the representation of (d). Define \(\rho^{\prime}\) by $$ \begin{array}{l} \rho^{\prime}(\sigma)=\rho(\sigma) \text { if } \sigma \text { is even: } \\ \rho^{\prime}(a)=-\rho(a) \text { if } \sigma \text { is odd. } \end{array} $$ Show that \(\rho^{\prime}\) is also irreducible, remains irreducible after tensoring with \(k^{\mathrm{a}}\), and is non-isomorphic to \(\rho\). This concludes the description of all irreducible representations of \(S_{4}\) (f) Show that the 3 -dimensional irreducible representations of \(S_{4}\) provide an irreducible representation of \(A_{4}\) (g) Show that all irreducible representations of \(A_{4}\) are given by the representations in (f) and three others which are one-dimensional.

The quaternion group. Let \(Q=\\{\pm 1, \pm x, \pm y, \pm z\\}\) be the quaternion group, with \(x^{2}=y^{2}=z^{2}=-1\) and \(x y=-y x, x z=-2 x, y z=-z y .\) (a) Show that \(Q\) has 5 conjugacy classes. Let \(A=\\{\pm 1\\}\). Then \(Q / A\) is of type \((2,2)\), and hence has 4 simple characters. which can be viewed as simple characters of \(Q\). (b) Show that there is only one more simple character of \(Q\), of dimension 2 . Show that the corresponding representation can be given by a matrix representation such that $$ \rho(x)=\left(\begin{array}{rr} i & 0 \\ 0 & -i \end{array}\right), \rho(y)=\left(\begin{array}{rr} 0 & 1 \\ -1 & 0 \end{array}\right), \rho(z)=\left(\begin{array}{ll} 0 & i \\ i & 0 \end{array}\right) $$ (c) Let \(\mathrm{H}\) be the quaternion field, i.e. the algebra over \(\mathrm{R}\) having dimension 4 , with basis \(\\{1, x, y, z\\}\) as in Exercise 3, and the corresponding relations as above. Show that \(\mathrm{C} \otimes_{\mathrm{R}} \mathrm{H}=\mathrm{Mat}_{2}(\mathrm{C})(2 \times 2\) complex matrices). Relate this to (b).

Let \(G\) be a finite group and \(S\) a normal subgroup. Let \(\rho\) be an irreducible representation of \(G\) over \(\mathrm{C}\). Prove that either the restriction of \(\rho\) to \(S\) has all its irreducible components \(S\) -isomorphic to cach other, or there exists a proper subgroup \(H\) of \(G\) containing \(S\) and an irreducible representation \(\theta\) of \(H\) such that \(\rho=\) ind \(C(\theta)\).

Dihedral group \(D_{2 m^{-}}\) There is a group of order \(2 n\) (n even integer \(\geqq 2\) ) generated by two elements \(\sigma, \tau\) such that $$ \sigma^{n}=1, \tau^{2}=1, \text { and } \tau \sigma \tau=\sigma^{-1} . $$ It is called the dihedral group. (a) Show that there are four representations of dimension 1 , obtained by the four possible values \(\pm 1\) for \(\sigma\) and \(\tau\). (b) Let \(C_{n}\) be the cyclic subgroup of \(D_{2 n}\) generated by \(\sigma\). For each integer \(r=0, \ldots, n-1\) let \(\psi\), be the character of \(C_{n}\) such that $$ \psi_{r}(\sigma)=\zeta^{r} \quad(\zeta=\text { prim. } n \text { -th root of unity }) $$ Let \(\chi_{r}\), be the induced character. Show that \(\chi_{r}=\chi_{n-r \cdot}\) (c) Show that for \(0

Let \(G\) be a finite group. semidirect product of \(A, H\) where \(A\) is commutative and normal. Let \(A^{\wedge}=\operatorname{Hom}\left(A, C^{*}\right)\) be the dual group. Let \(G\) operate by conjugation on characters, so that for \(\sigma \in G, a \in A\), we have $$ [\sigma] \psi(a)=\psi\left(\sigma^{-1} a \sigma\right) . $$ Let \(\psi_{1}, \ldots, \psi\), be representatives of the orbits of \(H\) in \(A^{\wedge}\), and let \(H_{i}(i=1, \ldots, r)\) be the isotropy group of \(\psi\). Let \(G_{i}=A H_{i}\). (a) For \(a \in A\) and \(h \in H_{i}\), define \(\psi_{i}(a h)=\psi_{i}(a) .\) Show that \(\psi_{i}\) is thus extended to a character on \(G_{i^{\circ}}\) Let \(\theta\) be a simple representation of \(H_{i}\) (on a vector space over C). From \(H_{i}=G_{i} / A\), view \(\theta\) as a simple representation of \(G_{i} .\) Let $$ \rho_{i, \theta}=\text { ind }_{G}^{G}\left(\psi_{i} \otimes \theta\right) . $$ (b) Show that \(\rho_{i, g}\) is simple. (c) Show that \(\rho_{i, \theta}=\rho_{i: y}\) implies \(i=i^{\prime}\) and \(\theta=\theta\). (d) Show that every irreducible representation of \(G\) is isomorphic to some \(\rho_{L . e}\)

Let \(G\) be a finite group operating on a finite set \(S\). Let \(\mathbf{C}[S]\) be the vector space generated by \(S\) over \(\mathbf{C}\). Let \(\psi\) be the character of the corresponding representation of \(G\) on \(\mathrm{C}[S]\). (a) Let \(\sigma \in G\). Show that \(\psi(\sigma)=\) number of fixed points of \(\sigma\) in \(S\). (b) Show that \(\left\langle\psi, 1_{G}\right\rangle_{G}\) is the number of \(G\) -orbits in \(S\).

Let \(A\) be a commutative subgroup of a finite group \(G .\) Show that every irreducible representation of \(G\) over \(\mathbf{C}\) has dimension \(\leqq(G: A)\).

Let \(\mathbf{F}\) be a finite field and let \(G=S L_{2}(\mathbf{F})\). Let \(B\) be the subgroup of \(G\) consisting of all matrices $$ \alpha=\left(\begin{array}{ll} a & b \\ 0 & d \end{array}\right) \in S L_{2}(\mathbf{F}), \text { so } d=a^{-1} $$ Let \(\mu: \mathbf{F}^{*} \rightarrow \mathbf{C}^{*}\) be a homomorphism and let \(\psi_{\mu}: B \rightarrow \mathbf{C}^{*}\) be the homomorphism such that \(\psi_{\mu}(\alpha)=\mu(a)\). Show that the induced character ind \(g\left(\psi_{\mu}\right)\) is simple if \(\mu^{2} \neq 1\)

Observe that \(A_{5} \approx S L_{2}\left(\mathbf{F}_{4}\right) \approx P S L_{2}\left(\mathbf{F}_{5}\right) .\) As a result, verify that there are 5 conjugacy classes, whose elements have orders 1, 2, 3, 5, 5 respectively, and write down explicitly the character table for \(A_{5}\) as was done in the text for \(G L_{2}\).

Let \(G\) be a \(p\) -group and let \(G \rightarrow \operatorname{Aut}(V)\) be a representation on a finite dimensional vector space over a field of characteristic \(p .\) Assume that the representation is irreducible. Show that the representation is trivial, i.e. \(G\) acts as the identity on \(V\).

Let \(G\) be a finite group and let \(C\) be a conjugacy class. Prove that the following two conditions are equivalent. They define what it means for the class to be rational. RAT 1. For all characters \(x\) of \(G, \chi(\sigma) \in \mathbf{Q}\) for \(\sigma \in C\). RAT 2\. For all \(\sigma \in C\), and \(j\) prime to the order of \(\sigma\), we have \(\sigma^{\prime} \in C\).

Let \(G\) be a group and let \(H_{1}, H_{2}\) be subgroups of finite index. Let \(\rho_{1}, \rho_{2}\) be representations of \(H_{1}, H_{2}\) on \(R\) -modules \(F_{1}, F_{2}\) respectively. Let \(M_{G}\left(F_{1}, F_{2}\right)\) be the \(R\) module of functions \(f: G \rightarrow \mathrm{Hom}_{R}\left(F_{1}, F_{2}\right)\) such that $$ f\left(h_{1} \sigma h_{2}\right)=\rho_{2}\left(h_{2}\right) f(\sigma) \rho_{1}\left(h_{1}\right) $$ for all \(\sigma \in G, h_{i} \in H_{i}(i=1,2) .\) Establish an \(R\) -module isomorphism $$ \operatorname{Hom}_{R}\left(F_{1}^{Q}, F_{\frac{T}{2}}^{G}\right) \rightarrow M_{G}\left(F_{1}, F_{2}\right) . $$ By \(F_{i}^{G}\) we have abbreviated ind \({ }_{A}^{G}\left(F_{i}\right)\).

(a) Let \(G_{1}, G_{2}\) be two finite groups with representations on \(\mathbf{C}\) -spaces \(E_{1}, E_{2}\). Let \(E_{1} \otimes E_{2}\) be the usual tensor product over \(\mathbf{C}\), but now prove that there is an action of \(G_{1} \times G_{2}\) on this tensor product such that $$ \left(\sigma_{1}, \sigma_{2}\right)(x \otimes y)=\sigma_{1} x \otimes \sigma_{2} y \text { for } \sigma_{1} \in G_{1}, \sigma_{2} \in G_{2} $$ This action is called the tensor product of the other two. If \(\rho_{1}, \rho_{2}\) are the representations of \(G_{1}, G_{2}\) on \(E_{1}, E_{2}\) respectively, then their tensor product is denoted by \(\rho_{1} \otimes \rho_{2}\). Prove: If \(\rho_{1}, \rho_{2}\) are irreducible then \(\rho_{2} \otimes \rho_{2}\) is also irreducible. [Hint: Use Theorem 5.17.] (b) Let \(\chi_{1}, \chi_{2}\) be the characters of \(\rho_{1}, \rho_{2}\) respectively. Show that \(x_{1} \otimes x_{2}\) is the character of the tensor product. By definition, $$ \chi_{1} \otimes x_{2}\left(\sigma_{1}, \sigma_{2}\right)=\chi_{1}\left(\sigma_{1}\right) \chi_{2}\left(\sigma_{2}\right) . $$

Let \(G\) be a finite set of endomorphisms of a finite-dimensional vector space \(E\) over the field \(k\). For each \(\sigma \in G\), let \(c\), be an element of \(k\). Show that if $$ \sum_{\sigma \in G} c_{\theta} T^{\prime}(\sigma)=0 $$ for all integers \(r \geqq 1\), then \(c_{e}=0\) for all \(\sigma \in G\). [Hint: Use the preceding exercise, and Proposition \(7.2\) of Chapter XVI. 1

(Steinberg). Let \(G\) be a finite monoid, and \(k[G]\) the monoid algebra over a field \(k\). Let \(G \rightarrow\) End \(_{2}(E)\) be a faithful representation (i.e. injective), so that we identify \(G\) with a multiplicative subset of End \(_{k}(E) .\) Show that \(T^{\prime}\) induces a representation of \(G\) on \(T^{\prime}(E)\). whence a representation of \(k[G]\) on \(T(E)\) by linearity. If \(\alpha \in k[G]\) and if \(T^{\prime}(\alpha)=0\) for all integers \(r \geq 1\), show that \(\alpha=0\). [Hint: Apply the preceding exercise.]

Let \(X(G)\) be the character ring of a finite group \(G\), generated over \(Z\) by the simple characters over \(\mathbf{C}\). Show that an element \(f \in X(G)\) is an effective irreducible character if and only if \((f, f)_{G}=1\) and \(f(1) \geq 0 .\)

In this exercise, we assume the next chapter on alternating produets. Let \(\rho\) be an irreducible representation of \(G\) on a vector space \(E\) over \(\mathbf{C}\). Then by functoriality we have the corresponding representations \(S^{\prime}(\rho)\) and \(\bigwedge^{\prime}(\rho)\) on the \(r\) -th symmetric power and \(r\) -th alternating power of \(E\) over \(\mathbf{C}\). If \(x\) is the character of \(\rho\), we let \(S^{\prime}(\chi)\) and \(\bigwedge^{\prime}(x)\) be the characters of \(S^{\prime}(\rho)\) and \(\bigwedge^{\prime}(\rho)\) respectively, on \(S^{\prime}(E)\) and \(\wedge^{\prime}(E)\). Let \(t\) be a variable and let $$ \sigma_{,}(\chi)=\sum_{r=0}^{\infty} S^{\prime}(x) t^{r}, \quad \lambda_{r}(x)=\sum_{r=0}^{\infty} \wedge^{\prime}(\chi) t^{r} . $$ (a) Comparing with Exercise 24 of Chapter XIV, prove that for \(x \in G\) we have $$ \sigma_{N}(x)(x)=\operatorname{det}(I-\rho(x) t)^{-1} \text { and } \lambda_{t}(x)(x)=\operatorname{det}(I+\rho(x) t) . $$ (b) For a function \(f\) on \(G\) define \(\Psi^{m}(f)\) by \(\Psi^{n}(f)(x)=f\left(x^{n}\right)\). Show that $$ -\frac{d}{d t} \log \sigma_{i}(x)=\sum_{n=1}^{\infty} \Psi^{n}(\chi) t^{n} \text { and }-\frac{d}{d t} \log \lambda_{-1}(\chi)=\sum_{n=1}^{\infty} \Psi^{n}(\chi) t^{n} $$ (c) Show that $$ n S^{n}(x)=\sum_{r=1}^{n} \Psi^{\prime}(\chi) S^{n-r}(x) \text { and } n \wedge^{n}(\chi)=\sum_{r=1}^{\infty}(-1)^{r-1} \Psi^{\prime}(\chi) \wedge^{n-\prime}(\chi) . $$

The following formalism is the analogue of Artin's formalism of \(L\) -series in number theory. Cf. Artin's "Zur Theorie der \(L\) -Reihen mit allgemeinen Gruppencharakteren", Collected papers, and also S. Lang. "L-series of a covering", Proc. Nat Acad. Sc. USA (1956). For the Artin formalism in a context of analysis, see J. Jorgenson and S. Lang, "Artin formalism and heat kernels", J. reine angew. Math. 447 (1994) pp. 165-200. We consider a category with objects \(\\{U\\}\). As usual, we say that a finite group \(\mathrm{G}\) operates on \(U\) if we are given a homomorphism \(\rho: G \rightarrow\) Aut \((U)\). We then say that \(U\) is a G-object, and also that \(\rho\) is a representation of \(G\) in \(U\). We say that \(G\) operates trivially if \(\rho(G)=\) id. For simplicity, we omit the \(\rho\) from the notation. By a G-morphism \(f: U \rightarrow V\) between \(G\) -objects, one means a morphism such that \(f \circ \sigma=\sigma \circ f\) for all \(\sigma \in G\). We shall assume that for each \(G\) -object \(U\) there exists an object \(U / G\) on which \(G\) operates trivially, and a \(G\) -morphism \(\pi_{v, G}: U \rightarrow U / G\) having the following universal property: If \(f: U \rightarrow U^{\prime}\) is a \(G\) -morphism, then there exists a unique morphism $$ f / G: U / G \rightarrow U^{\prime} / G $$ making the following diagram commutative: In particular, if \(H\) is a normal subgroup of \(G\), show that \(G / H\) operates in a natural way on \(U / H\). Let \(k\) be an algebraically closed field of characteristic \(0 .\) We assume given a functor \(E\) from our category to the category of finite dimensional \(k\) -spaces. If \(U\) is an object in our category, and \(f: U \rightarrow U^{\prime}\) is a morphism, then we get a homomorphism $$ E(f)=f_{*}: E(U) \rightarrow E(U) $$ (The reader may keep in mind the special case when we deal with the category of reasonable topological spaces, and \(E\) is the homology functor in a given dimension.) If \(G\) operates on \(U\), then we get an operation of \(G\) on \(E(U)\) by functoriality. Let \(U\) be a \(G\) -object, and \(F: U \rightarrow U\) a \(G\) -morphism. If \(P_{r}(t)=\Pi\left(t-a_{1}\right)\) is the characteristic polynomial of the linear map \(F_{t}: E(U) \rightarrow E(U)\), we define $$ Z_{f}(t)=\prod\left(1-\alpha_{i} t\right) $$ and call this the zeta function of \(F\). If \(F\) is the identity, then \(Z_{P}(t)=(1-t)^{\text {R } W}\) ) where we define \(B(U)\) to be \(\operatorname{dim}, E(U)\) Let \(\chi\) be a simple character of \(G\). Let \(d_{z}\) be the dimension of the simple representation of \(G\) belonging to \(\chi\), and \(n=\operatorname{ord}(G) .\) We define a linear map on \(E(U)\) by letting $$ e_{x}=\frac{d_{x}}{n} \sum_{\text {eff } G} \chi\left(\sigma^{-1}\right) \sigma_{\text {* }} $$ If \(P_{x}(t)=\prod\left(t-\beta_{X}(x)\right)\) is the characteristic polynomial of \(e_{x}=F_{*}\), define $$ L_{p}(t, \chi, U / G)=\prod(1-\beta(x) t) $$ Show that the logarithmic derivative of this function is equal to $$ -\frac{1}{N} \sum_{\mu=1}^{\infty} \operatorname{tr}\left(e_{x}=F_{\%}^{\mu}\right) r^{\mu-1} $$ Define \(L_{p}(t, \chi, U / G)\) for any character \(x\) by linearity. If we write \(V=U / G\) by abuse of notation, then we also write \(L_{p}(t, \chi, U / V)\). Then for any \(\chi, \chi\) ' we have by definition, $$ L_{P}\left(t_{+} x+\chi^{\prime}, U / V\right)=L_{F}(t, \chi, U / V) L_{F}\left(t, X^{\prime}, U / V\right) $$ We make one additional assumption on the situation: Assume that the characteristic polynomial of $$ \frac{1}{n} \sum_{d \in G} \sigma_{*} \cdot F_{*} $$ is equal to the characteristic polynomial of \(F / G\) on \(E(U / G)\). Prove the following statement: (a) If \(G=\\{1\\}\) then $$ L_{F}(t, 1, U / U)=Z_{p}(t) $$ (b) Let \(V=U / G\). Then $$ L_{p}(t, 1, U / V)=Z_{F}(t) $$ (c) Let \(H\) be a subgroup of \(G\) and let \(\psi\) be a character of \(H\). Let \(W=U / H\), and let \(\psi^{\theta}\) be the induced character from \(H\) to \(G\). Then $$ L_{F}(t, \psi, U / W)=L_{F}\left(t, \psi^{G}, U / V\right) $$ (d) Let \(H\) be normal in \(G\). Then \(G / H\) operates on \(U / H=W\). Let \(\psi\) be a character of \(G / H\), and let \(\gamma\) be the character of \(G\) obtained by composing \(\psi\) with the canonical map \(G \rightarrow G / H .\) Let \(\varphi=F / H\) be the morphism indused on $$ U / H=W $$ Then $$ L_{\varphi}(t, \psi, W / V)=L_{F}(t, \chi, U / V) $$ (e) If \(V=U / G\) and \(B(V)=\operatorname{dim}_{k} E(V)\), show that \((1-t)^{n \cdot \eta_{1}}\) divides \((1-t)^{\text {At }}\). Use the regular character to determine a factorization of \((1-t)^{\operatorname{sic}}\).