Brian Bi
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Problem 5.1.2

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 $$|G| = 1$$.