Brian Bi
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Problem 5.1.2

  1. We follow the Hint.

    Lemma 1: \(V_\mathbb{C} \cong V \oplus V^*\).

    The complexification of \(V\) is defined as \(V_{\mathbb{C}} = \mathbb{C} \otimes_{\mathbb{R}} V\) where elements of \(G\) act on the second factor and elements of \(\mathbb{C}\) act on the first. It is clear that \(V_{\mathbb{C}} = 1 \otimes_{\mathbb{R}} V \oplus i \otimes_{\mathbb{R}} V\). We can write \((v, w)\) for the element \(1 \otimes v + i \otimes w\). It's clear that \(x(v, w) = (xv, xw)\) when \(x \in \mathbb{R}[G]\), and \(i(v, w) = (-w, v)\).

    The representation \(V_\mathbb{C}\) has two invariant subspaces \(V_+ = \{(v, iv) \mid v \in V\}\) and \(V_- = \{(v, -iv) \mid v \in V\}\). It is not hard to see that \(V_{\mathbb{C}} = V_+ \oplus V_-\). We have an isomorphism of vector spaces \(V \to V_-\) given by \(\varphi_-(v) = (v, -iv)\). This is also an isomorphism of representations since for all \(x \in \mathbb{C}[G]\), we have \begin{align*} \varphi_-(xv) &= (xv, -ixv) \\ &= (x_r v + ix_i v, -i(x_r v + ix_iv)) \\ &= (x_r v, -ix_r v) + (i x_i v, x_i v) \\ &= x_r(v, -iv) + ix_i(v, -iv) \\ &= x\varphi_-(v) \end{align*} where \(x_r = \Re(x), x_i = \Im(x)\). There is also an isomorphism of representations \(\varphi_+ : \overline{V} \to V_+\) given by \(\varphi_+(v) = (v, iv)\); we verify this below: \begin{align*} \varphi_+(\rho_{\overline{V}}(x)v) &= \varphi_+(\rho_V(\overline{x})v) \\ &= (\rho_V(\overline{x})v, i\rho_V(\overline{x})v) \\ &= (x_r v - ix_i v, i(x_r v - ix_i v)) \\ &= (x_r v, ix_r v) + (-ix_i v, x_i v) \\ &= x_r(v, iv) + ix_i(v, iv) \\ &= x\varphi_+(v) \end{align*} Therefore \(V_\mathbb{C} \cong V \oplus \overline{V} \cong V \oplus V^*\).

    Lemma 2: \(\End_{\mathbb{C}[G]}(V_\mathbb{C})\) and \(\End_{\mathbb{R}[G]}(V)^2\) are isomorphic as \(\mathbb{R}\)-vector spaces.

    Suppose \(\varphi_1, \varphi_2 \in \End_{\mathbb{R}[G]}(V)\). We claim that there is a unique \(\varphi \in \End_{\mathbb{C}[G]}\) such that \(\pi_1(\varphi(v, 0)) = \varphi_1(v)\) and \(\pi_2(\varphi(v, 0)) = \varphi_2(v)\) for all \(v \in V\). We will first show that \(\varphi\) is unique if it exists. Let \(v \in V\); then \(\varphi(0, v) = \varphi(i(v, 0)) = i\varphi(v, 0) = i(\varphi_1(v), \varphi_2(v)) = (-\varphi_2(v), \varphi_1(v))\). By linearity, \(\varphi\) is completely determined. Now we will show that the \(\varphi\) so constructed is indeed an endomorphism. It is clear that the formula for \(\varphi\) is \(\varphi(v, w) = (\varphi_1(v) - \varphi_2(w), \varphi_2(v) + \varphi_1(w))\). Let \(x \in \mathbb{C}[G]\), with real part \(x_r\) and imaginary part \(x_i\). Then, \begin{align*} \varphi(x(v, w)) &= \varphi(x_r v - x_i w, x_r w + x_i v) \\ &= (\varphi_1(x_r v - x_i w) - \varphi_2(x_r w + x_i v), \varphi_2(x_r v - x_i w) + \varphi_1(x_r w + x_i v)) \\ &= (x_r\varphi_1(v) - x_i \varphi_1(w) - x_r \varphi_2(w) - x_i \varphi_2(v), x_r \varphi_2(v) - x_i \varphi_2(w) + x_r\varphi_1(w) + x_i \varphi_1(v)) \\ &= (x_r(\varphi_1(v) - \varphi_2(w)) - x_i(\varphi_2(v) + \varphi_1(w)), x_r(\varphi_2(v) + \varphi_1(w)) + x_i(\varphi_1(v) - \varphi_2(w))) \\ &= x (\varphi_1(v) - \varphi_2(w), \varphi_2(v) + \varphi_1(w)) \\ &= x \varphi(v, w) \end{align*} therefore \(\varphi\) is indeed an endomorphism of \(V_\mathbb{C}\).

    Conversely, if \(\varphi \in \End_{\mathbb{C}[G]}(V_{\mathbb{C}})\), then we claim that the functions \(\varphi_1(v) = \pi_1(\varphi(v, 0))\) and \(\varphi_2(v) = \pi_2(\varphi(v, 0))\) are endomorphisms of \(V\) as a \(\mathbb{R}[G]\)-module. To see this, let \(x \in \mathbb{R}[G]\) and observe that \(\varphi_1(xv) = \pi_1(\varphi(xv, 0)) = \pi_1(x\varphi(v, 0)) = x\pi_1(\varphi(v, 0)) = x\varphi_1(v)\), and similarly with \(\varphi_2\).

    Thus, there is a bijective correspondence between \(\End_{\mathbb{C}[G]}(V_{\mathbb{C}})\) and \(\End_{\mathbb{R}[G]}(V)^2\). As we have defined it, it is also clear that it is \(\mathbb{R}\)-linear.

    Corollary: \(\dim_{\mathbb{C}} \End_{\mathbb{C}[G]}(V_{\mathbb{C}}) = \frac{1}{2} \dim_{\mathbb{R}} \End_{\mathbb{C}[G]}(V_{\mathbb{C}}) = \dim_{\mathbb{R}} \End_{\mathbb{R}[G]}(V)\).

    We now return to the problem. Lemma 1 gives us the decomposition \(V_{\mathbb{C}} \cong V \oplus V^*\). The representation \(V\) was given to be irreducible, so \(V^*\) is also irreducible. If \(V \not\cong V^*\), then Schur's lemma for algebraically closed fields implies that \(\End_{\mathbb{C}[G]}(V_{\mathbb{C}})\) is parametrized by \(z, z' \in \mathbb{C}\), the factors by which the endomorphism scales the \(V\) and \(V^*\) components in \(V_{\mathbb{C}}\). Therefore we have \(\dim_{\mathbb{C}} \End_{\mathbb{C}[G]}(V_{\mathbb{C}}) = 2\), and by the Corollary, \(\dim \End_{\mathbb{R}[G]}(V) = 2\). In the case where \(V \cong V^*\), Schur's lemma implies that \(\dim_{\mathbb{C}} \End_{\mathbb{C}[G]}(V_{\mathbb{C}}) = 4\), as such endomorphisms are parametrized by four complex numbers, each of which is the scaling factor from one of the two copies of \(V\) to one of the two copies of \(V\). Therefore, by the Corollary, \(\dim \End_{\mathbb{R}[G]}(V) = 4\).

    Now, every endomorphism of \(V\) as a \(\mathbb{C}[G]\)-module is also an endomorphism of \(V\) as an \(\mathbb{R}[G]\)-module, therefore \(\mathbb{C} \subseteq \End_{\mathbb{R}[G]}(V)\). In the complex case (\(V \not\cong V^*\)), since \(\dim \End_{\mathbb{R}[G]} = 2\) and \(\dim_{\mathbb{R}}\mathbb{C} = 2\), we conclude that in fact \(\mathbb{C} = \End_{\mathbb{R}[G]}(V)\).

    We now have to distinguish between the real and quaternionic cases.

    Let \(\langle \cdot, \cdot \rangle\) be the invariant positive Hermitian form on \(V\), which is unique up to a positive real scaling factor by Theorem 4.6.2. (See here for a refresher.)

    A similar argument shows that the invariant nondegenerate form \(B\) described in Definition 5.1.1 is unique up to a nonzero complex scaling factor. By nondegeneracy, there is a unique \(j \in \operatorname{Aut}_{\mathbb{R}}(V)\) such that \(B(v, w) = \langle v, j(w)\rangle\) for all \(v, w \in V\).

    Let us investigate the properties of \(j\). First, \[ B(v, iw) = iB(v, w) = i\langle v, j(w)\rangle = \langle v, -ij(w)\rangle \] so \(j(iw) = -ij(w)\); in other words, \(j\) is antilinear. In addition, for each \(g \in G\), \[ B(v, gw) = B(g^{-1}v, w) = \langle g^{-1}v, j(w)\rangle = \langle v, gj(w) \rangle \] so \(j(gw) = gj(w)\). It follows that \(j \in \operatorname{Aut}_{\mathbb{R}[G]}(V)\), and if \(x \in \mathbb{C}[G]\), then \(j(xw) = \overline{x}(jw)\). Since \(j\) is antilinear, \(j^2\) is linear, and \(j^2(xw) = j(\overline{x}j(w)) = xj^2(w)\), so \(j^2 \in \operatorname{Aut}_{\mathbb{C}[G]}(V)\). By Schur's lemma, \(j^2 = \lambda \Id\) for some nonzero \(\lambda \in \mathbb{C}\).

    Choose some nonzero \(v \in V\). Then \[ \overline{\lambda} \langle v, v \rangle = \langle v, j^2(v) \rangle = B(v, j(v)) = \pm B(j(v), v) = \pm \langle j(v), j(v)\rangle \] where the plus sign is for \(B\) symmetric and the minus sign for \(B\) skew-symmetric. Write \(\overline{\lambda} = \pm \langle j(v), j(v)\rangle/\langle v, v \rangle\). Since \(\langle v, v\rangle\) and \(\langle j(v), j(v)\rangle\) are both positive real, it follows that \(\lambda\) is positive real for \(B\) symmetric (\(V\) real) and negative real for \(B\) skew-symmetric (\(V\) quaternionic). In the real case, we can rescale \(B\) by \(\lambda^{-1/2}\) so that \(j^2 = 1\), and in the quaternionic case, we can rescale \(B\) by \((-\lambda)^{-1/2}\) so that \(j^2 = -1\); we will assume that this has been done.

    We know that \(\End_{\mathbb{R}[G]}(V)\) contains a copy of \(\mathbb{C}\), but since \(j\) is nonzero antilinear, \(j\) is not contained within \(\mathbb{C}\). For each \(z \in \mathbb{C}\), \(zj \in \End_{\mathbb{R}[G]}(V)\), and if \(z\) is nonzero, then \(zj\) is also nonzero antilinear, so it is also not contained in \(\mathbb{C}\). Since \(\mathbb{C}\) and \(\mathbb{C}j\) have trivial intersection and each has (real) dimension 2, and \(\End_{\mathbb{R}[G]}(V)\) has dimension 4, it follows that \(\End_{\mathbb{R}[G]}(V) = \mathbb{C} \oplus \mathbb{C}j\) (as vector spaces). This implies that the rules \(j^2 = \pm 1\), \(ji = -ij\) completely determine the isomorphism class of the algebra \(\End_{\mathbb{R}[G]}(V)\); this algebra is isomorphic to any 4-dimensional real associative algebra \(A\) such that \(A\) contains a copy of the complex numbers and an element \(J\) such that \(J^2 = \pm 1\) and \(Ji = -iJ\).

    In the quaternionic case, where \(j^2 = -1\), the quaternions \(\mathbb{H}\) form an algebra with the above properties (where we take \(J\) to be the element \(j\), although \(j \cos \theta + k \sin \theta\) would also do for any \(\theta\)). Thus \(\End_{\mathbb{R}[G]}(V) \cong \mathbb{H}\). In the real case, we can take \(A = \operatorname{Mat}_2(\mathbb{R})\). To see this, recall the well-known fact that \(A\) contains a copy of the complex numbers, with the correspondence \[ x + yi \longleftrightarrow \begin{pmatrix} x & y \\ -y & x \end{pmatrix} \] and we can take \[ J = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \] whereupon \[ Ji = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = -\begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} = -\begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} = -iJ \] as claimed.

  2. First, let \(V_\mathbb{R}\) be a real representation. Choose \(B\) to be a positive definite symmetric bilinear form on \(V_{\mathbb{R}}\). The construction of Theorem 4.6.2 then gives an invariant positive definite symmetric form \(\overline{B}\), which is automatically nondegenerate. On the complexification \(V\), define \[ \overline{B}(v + iw, v' + iw') = \overline{B}(v, v') - \overline{B}(w, w') + i(\overline{B}(w, v') + \overline{B}(v, w')) \] This is a symmetric \(G\)-invariant \(\mathbb{C}\)-bilinear form on \(V\) (we omit the proof since it is straightforward but tedious). Additionally, if \(v, w\) are fixed and at least one is nonzero, there is always \(v', w'\) such that \(\overline{B}(v + iw, v' + iw') \ne 0\). To wit, if \(v\) is nonzero we could for example choose \(v'\) so that \(\overline{B}(v, v')\) is nonzero and set \(w' = 0\), or if \(w\) is nonzero then we could choose \(w'\) so that \(\overline{B}(w, w')\) is nonzero and set \(v' = 0\). Therefore \(\overline{B}\) is a symmetric \(G\)-invariant \(\mathbb{C}\)-bilinear form on \(V\), and \(V\) is of real type.

    Conversely, suppose \(V\) is of real type. Let \(B\) be the nondegenerate \(G\)-invariant symmetric \(\mathbb{C}\)-bilinear form on \(V\). There exists \(v \in V\) such that \(B(v, v) \ne 0\), and we may normalize \(v\) so that \(B(v, v) = 1\); we assume this has been done. Let \(V_{\mathbb{R}} = \mathbb{R}[G]v\). \(G\)-invariance implies that \(B(gv, gv) = 1\) for all \(g \in G\), and bilinearity then implies that \(B(w, w)\) is real and positive for all nonzero \(w \in V_{\mathbb{R}}\) and that \(B(w, w)\) is real and negative for all nonzero \(w \in iV_{\mathbb{R}}\). Therefore \(V_{\mathbb{R}} \cap iV_{\mathbb{R}} = \{0\}\). But \(V_{\mathbb{R}} + iV_{\mathbb{R}} = \mathbb{R}[G]v + i\mathbb{R}[G]v = \mathbb{C}[G]v\), which is all of \(V\) since \(V\) is irreducible. Therefore, \(V = V_{\mathbb{R}} \oplus iV_{\mathbb{R}}\) where the two components are \(\mathbb{R}[G]\)-modules; that is, \(V\) is the complexification of \(V_{\mathbb{R}}\).

Remark: Etingof's proof of Theorem 5.1.5 (the Frobenius–Schur involution formula) leaves out some steps.

First, the elements of \(S^2 V\) are the symmetric bilinear forms on \(V^*\), and for all \(g \in G, B \in S^2 V\), we have \(B(g\varphi_1, g\varphi_2) = g^{-1}B(\varphi_1, \varphi_2)\). Therefore \(B\) is \(G\)-invariant on \(V^*\) if and only if \(gB = B\) for all \(g \in G\). If such nonzero \(B\) exists, it is necessarily nondegenerate (the proof is left as an exercise) and therefore unique up to a scaling factor. Thus, \(\dim (S^2 V)^G\) is 1 when \(V^*\) has a nondegenerate \(G\)-invariant symmetric bilinear form, and 0 otherwise. But \(V^*\) is real if and only if \(V\) is real, which implies that \(\dim (S^2 V)^G\) is 1 when \(V\) is real and 0 otherwise. A similar argument shows that \(\dim (\wedge^2 V)^G\) is 1 when \(V\) is quaternionic and 0 otherwise. This establishes the claim that \(FS(V) = |G|^{-1}\sum_{g\in G} \chi_V(g^2)\).

Second, Etingof immediately concludes that the number of involutions in \(G\) is \(\sum_V \dim V\ FS(V)\), which is not at all obvious. A proof of this fact is given here (section 2.4). The key is to recognize that the function \(\theta : G \to \mathbb{N}\) where \(\theta(g)\) is the number of elements that square to \(g\) is a class function, and can therefore be written as \(\theta = \sum_V \chi_V \langle \theta, \chi_V\rangle\). Some computation then gives the desired result.

Exercise 5.1.7 Suppose \(|G|\) is odd and all representations of \(\mathbb{C}[G]\) are real. Then \(G\) has no elements of order 2, so its only involution is the identity. By Corollary 5.1.6, the sum of the dimensions of all irreps of \(G\) is 1. This implies that \(G\) has only one irrep, with dimension 1 (namely the trivial representation). Maschke's theorem then implies that \(|G| = 1\).