7  Characters

The character of a group representation \(\rho: G \to \operatorname{GL}(V)\) is a function on \(G\) associating to each group element the trace of the corresponding matrix, i.e., the sum of its eigenvalues. As such, the character carries essential information about the representation in highly condensed form, motivated by the key role that eigenvalues took in our ad hoc approaches to computing irreducible decompositions.

While we cannot generally recover the eigenvalues of a matrix from its trace, we have seen that we can recover the characteristic polynomial for each element \(\rho(g)\) given the data of the entire representation (recall Equation 6.1). This leads us, morally, to investigate the theory of characters.

7.1 Introduction

Definition 7.1 The character of a representation \(\rho: G \to \operatorname{GL}(V)\) is the complex-valued function \(\chi_V\) on the group defined by \(\chi_V(g) \coloneqq \operatorname{Tr}\rho(g)\).

Example 7.1 The character of the trivial representation is the constant function \(1\). More generally, the character of a 1-dimensional representation is the representation itself.

For the next example, recall Euler’s formula \(e^{ix} = \cos x + i \sin x\) and its rephrasings: \[ \cos x = \frac{e^{ix} + e^{-ix}}{2} \quad \text{and} \quad \sin x = \frac{e^{ix} - e^{-ix}}{2i}. \]

Example 7.2 The character of the dihedral action on the plane (Example 4.7) is given by \[ \chi(r^k) = 2 \cos\left( \tfrac{2k\pi}{n} \right) \quad \text{ and } \quad \chi(s) = 0. \]

Example 7.3 Recall the reducible but not decomposable action of Example 4.17, whose character is the constant function \(2\).

Proposition 7.1 Let \(\chi\) be the character of \(\rho: G \to \operatorname{GL}(V)\) with \(G\) finite. Then \[ \ker \rho = \left\{ g \in G: \chi(g) = \dim V \right\}. \]

For this reason, we also call \(\ker \rho\) the kernel of \(\chi\).

Proof. If \(g \in \ker \rho\), then \(\chi(g) = \operatorname{Tr}(\rho(g)) = \operatorname{Tr}(\mathbb{I}) = \dim V\). Conversely, by Lemma Lemma 4.1, the eigenvalues \(\lambda_1, \dots, \lambda_n\) of each \(\rho(g)\) are norm \(1\) complex numbers. By the triangle inequality, \[ |\chi(g)| = |\lambda_1 + \cdots + \lambda_n| \leq |\lambda_1| + \cdots + |\lambda_n| = n, \] with equality if and only if each \(\lambda_i\) have the same argument. In particular, we can only have \(\chi(g) = \dim V\) if each eigenvalue is \(1\), i.e., if \(\rho(g) = \mathbb{I}\).

First we verify that characters are an isomorphism invariant:

Proposition 7.2 If \(V\) and \(W\) are isomorphic \(G\)-representations, then \(\chi_V = \chi_W\).

Proof. If \(\rho: G \to \operatorname{GL}(V)\) and \(\sigma: G \to \operatorname{GL}(W)\) and \(\varphi: V \to W\) is an isomorphism of \(G\)-representations, then \(\varphi \circ \rho(g) = \sigma(g) \circ \varphi\) for each \(g \in G\). Thus \(\rho(g) = \varphi^{-1} \circ \sigma(g) \circ \varphi\) and \[ \chi_V(g) = \operatorname{Tr}( \rho(g) ) = \operatorname{Tr}( \varphi^{-1} \circ \sigma(g) \circ \varphi ) = \operatorname{Tr}( \sigma(g) \circ \varphi \circ \varphi^{-1} ) = \operatorname{Tr}( \sigma(g) ) = \chi_W(g). \]

Moreover, in light of Proposition 6.1, Equation 6.4, Proposition 6.3, we can compute the characters of constructed representations in terms of their constituents:

Proposition 7.3 Let \(V\) and \(W\) be complex representations of a finite group \(G\). Then \[ \begin{array}{c} \chi_{V \oplus W} = \chi_V + \chi_W, \\ \chi_{V^*} = \overline{\chi_V}, \\ \chi_{V \otimes W} = \chi_V \chi_W, \\ \chi_{\operatorname{Hom}(V,W)} = \overline{\chi_V} \chi_W. \end{array} \]

Proof. The only trace we have not explicitly computed in the results referenced above is that on \(\operatorname{Hom}(V,W)\). In light of the isomorphism of \(G\)-represenations (Equation 6.8), we have \[ \chi_{\operatorname{Hom}(V,W)} = \chi_{V^*\otimes W} = \overline{\chi_V} \chi_W. \]

Students who have already brushed with topics in complex analysis, Hilbert spaces, or quantum mechanics might be suspicious (and rightfully so) of the right handside of the above expression. More on this momentarily!

We also restate a result proved in Homework (Exercise A.6) in the language of characters:

Proposition 7.4 If \(V\) is the permutation representation of the action of \(G\) on a finite set \(X\), then \(\chi_V(g)\) is the number of elements of \(X\) fixed by \(g\).

Example 7.4 Recall the standard representation \(V\) of \(\mathcal{S}_{3}\) (Example 6.1), given by the orthogonal complement of \(\operatorname{Span}\{e_1+e_2+e_3\}\). We can easily deduce the character of \(V\) using the fact that \(\mathbb{C}^3 = U \oplus V\): \[ \chi_V = \chi_{\mathbb{C}^3} - \chi_U = \chi_{\mathbb{C}^3} - 1. \] For example, \(\chi_V((1\ 2)) = \chi_{\mathbb{C}^3}((1\ 2)) - 1 = 1 - 1 = 0\), because \((1\ 2) \in \mathcal{S}_{3}\) fixes only \(3\).

7.2 Fixed Subspace

We began this section with the goal in mind to determine the irreducible representations of a given finite group, along with a technique for deducing the constituent irreducibles for an arbitrary representation. We shall take a step towards that goal by first learning to measure the fixed subspace, \(V^G\), of a given representation \(V\). To do so, we return to the spirit which animated Weyl’s unitary trick, i.e., the map given by \[ \varphi \coloneqq \frac{1}{|G|} \sum_{h \in G} \rho(h): V \to V. \] In particular, recall (Theorem 5.2) that \(\varphi\) is a projection onto \(V^G\). We recall the proof here:

Proof. We will show that \(\operatorname{im}\varphi \subseteq V^G\) and that \(\varphi|_{V^G} = \mathbb{I}|_{V^G}\), from which it follows that \(\varphi^2 = \varphi\). Towards that end, take \(x \in V, y \in V^G\), and \(g \in G\). Then we have: \[ \begin{split} g \cdot \varphi(x) & = g \cdot \left( \frac{1}{|G|} \sum_{h \in G} h \cdot x \right) = \frac{1}{|G|} \sum_{h \in G} \overbrace{(gh) \cdot x = \frac{1}{|G|} \sum_{h' \in G} h'}^{\text{re-index } h'=gh} \cdot x = \varphi(x), \\ \varphi(y) & = \frac{1}{|G|} \sum_{h \in G} h \cdot y = \frac{1}{|G|} \sum_{h \in G} y = y. \end{split} \]

Taking the trace of \(\varphi\), with a nod to Theorem 2.17 (also Exercise B.1, Exercise D.6), yields:

Corollary 7.1 If \(G\) is finite with representation \(G \to \operatorname{GL}(V)\) and trace \(\chi: G \to \mathbb{C}\), then \[ \dim V^G = \frac{1}{|G|} \sum_{g \in G} \chi(g). \]

Example 7.5 Returning to the permutation action of \(\mathcal{S}_{3}\) on \(\mathbb{C}^3\), the projection operator of Theorem 5.2 is given by an average of permutation matrices: \[ \varphi = \frac{1}{6} \left( \mathbb{I}+ \begin{psmallmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{psmallmatrix} + \begin{psmallmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{psmallmatrix} + \begin{psmallmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{psmallmatrix} + \begin{psmallmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{psmallmatrix} + \begin{psmallmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{psmallmatrix} \right) = \frac{1}{3} \begin{psmallmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{psmallmatrix} \] Note that \(\operatorname{Tr}\varphi = 1 = \dim (\mathbb{C}^3)^{\mathcal{S}_{3}}\) and the image of \(\varphi\) is spanned by the eigenvector \(e_1+e_2+e_3\) (read off from, say, the first column). This agrees with our analysis in Example 4.16, where we learned about the trivial and standard representations.

Example 7.6 Recall the action \(\mathcal{D}_{2n} ⟳\mathbb{R}^2\) with \(\chi(r^k) = 2 \cos(k\tfrac{2\pi}{n})\) and \(\chi(sr^k) = 0\). The average of characters as in Corollary 7.1 gives \[ \underbrace{1+\cos(\tfrac{2\pi}{n}) + \cdots + \cos((n-1)\tfrac{2\pi}{2})}_{\text{rotations}} + \underbrace{0 + 0 + \cdots + 0}_{\text{flips}} = 0. \] One way to see that these terms add to zero is to note that \(\cos(k \tfrac{2\pi}{n})\) is the \(x\)-coordinate of the \(k\)th vertex of a regular \(n\)-gon with vertices on the unit circle: the sum is therefore proportional to the average \(x\)-coordinate, i.e., \(0\). Another way to see the result is that the sum is equal to the real component of \(1+\zeta+\cdots+\zeta^{n-1} = 0\), where \(\zeta\) is a primitive \(n\)th root of unity. No matter which way we see it, the average value over the character is zero and so this representation has no nontrivial invariant vectors.

Recall the cyclic permutation property of traces, that \(\operatorname{Tr}( A B ) = \operatorname{Tr}( B A )\). In particular, this means that characters are constant on conjugacy classes of \(G\): \[ \chi_V(h g h^{-1}) = \operatorname{Tr}\left( \rho(h) \rho(g) \rho(h^{-1}) \right) = \operatorname{Tr}\rho(g) = \chi_V(g). \] Such functions are known as the \(\mathbb{C}\)-valued class functions on \(G\); we will denote the set of all such functions as \(\mathscr{C}({G})\). We can see this, for example, in Example 7.5: the trace is \(1\) when the underlying permutation is a \(2\)-cycle but \(0\) for \(3\)-cycles (recall Example 3.20).

When averaging a class function \(\chi: G \to \mathbb{C}\) over every element in the group, we can save ourselves some work by only checking the value of \(\chi\) on representatives of each conjugacy class, then multiplying by the size of said class. If we write \(g_1 = e, g_2, \cdots, g_r\) as representatives for the conjugacy classes of \(G\), then we can reformulate Corollary 7.1 as \[ \dim V^G = \frac{1}{|G|} \sum_{i=1}^r |\operatorname{cl}_{G}({g_i})|\ \chi(g_i). \tag{7.1}\] Before proceeding, recall the notion of indicator functions; for any \(A \subseteq G\), we set \[ \begin{aligned} \mathbb{1}_{A}: G & \to \mathbb{C}^\times \\ g & \mapsto \left\{ \begin{array}{ll} 1 & \text{if } g \in A \\ 0 & \text{otherwise.} \end{array} \right. \end{aligned} \] The functions \(\mathbb{1}_{\operatorname{cl}_{}({g_i})}: G \to \mathbb{C}\) give a basis for \(\mathscr{C}({G})\), so \(\dim \mathscr{C}({G}) = r\).

7.3 Orthonormality

We have seen from Proposition 5.3 that the space of intertwiners between representations is highly constrained by the irreducible substructure of those representations. Indeed, if \(V\) and \(W\) are representations and \(V_1, \dots, V_r\) are the irreducible representations of \(G\), then bases respecting the decompositions \[ V = V_1^{\oplus a_1} \oplus \cdots \oplus V_r^{\oplus a_r} \quad \text{ and } \quad W = V_1^{\oplus b_1} \oplus \cdots \oplus V_r^{\oplus b_r} \] lead to intertwiner matrices whose blocks correspond to maps between isotypic components, \(V_i^{\oplus a_i} \to V_j^{\oplus b_j}\), which are highly constrained by Schur’s Lemma.

At the same time, by Theorem 6.2 the condition of \(\varphi \in \operatorname{Hom}(V,W)\) being \(G\)-linear is the same as being fixed by the induced \(G\)-action. In this way, computing the dimension of \(G\)-invariant linear maps \(V \to W\) relates to identifying common irreducibles of \(V\) and \(W\).

Example 7.7 Recall Example 5.5, Example 5.6. In the standard basis \(E_{11}, E_{12}, \cdots, E_{33}\) for \(\operatorname{Hom}(\mathbb{C}^3_\rho,\mathbb{C}^3_\sigma)\), where \(E_{ij}\) is the matrix with \(1\) in the \((i,j)\) entry and \(0\) elsewhere, we can write down the induced \(\mathcal{S}_{3}\)-action: \[ \begin{split} (1\ 2) & \mapsto \begin{psmallmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \end{psmallmatrix}, \\ (1\ 2\ 3) & \mapsto \begin{psmallmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\zeta & -\zeta & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \zeta & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & \zeta^2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\zeta^2 & -\zeta^2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \zeta^2 & 0 \end{psmallmatrix}. \end{split} \] Using Equation 7.1, we can compute the dimension of the invariant subspace via \[ \dim \operatorname{Hom}(\mathbb{C}^3_\rho,\mathbb{C}^3_\sigma)^{\mathcal{S}_{3}} = \frac{1}{6} (1 \cdot 9 + 3 \cdot 1 + 2 \cdot 0) = 2, \] where \(9 = \operatorname{Tr}(\mathbb{I})\) and \(1\) and \(0\) are the traces of the above matrices (respectively). This calculation reflects the fact that these representations have two irreducibles in common, \(\mathbb{C}^3_\rho \cong U \oplus V \cong \mathbb{C}^3_\sigma\), with \(U\) the trivial representation and \(V\) the standard representation, as we will see developed more thoroughly in this section.

Similarly, for the action on \(\operatorname{Hom}(\mathbb{C}^3_\tau,\mathbb{C}^3_\sigma)\) we have \[ \small (1\ 2) \mapsto \begin{psmallmatrix} -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & -1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & -1 \\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & -1 & 0 & 0 & 0 \end{psmallmatrix}, \] with the action of \((1\;2\;3)\) as before, and so \[ \dim \operatorname{Hom}(\mathbb{C}^3_\tau,\mathbb{C}^3_\sigma)^{\mathcal{S}_{3}} = \tfrac{1}{6} (1 \cdot 9 + 3 \cdot (-1) + 2 \cdot 0) = 1. \] This reflects \(\mathbb{C}^3_\tau\) only having one irreducible, namely \(V\), in common with \(\mathbb{C}^3_\sigma\).

Examples like this and our associated insights lead us to the following:

Theorem 7.1 (Orthonormality of characters) Let \(G\) be a finite group with \(V\) and \(W\) irreducible representations over \(\mathbb{C}\) with characters \(\chi_V\) and \(\chi_W\), respectively. Then \[ \frac{1}{|G|} \sum_{g \in G} \overline{\chi_V(g)} \chi_W(g) = \left\{ \begin{array}{ll} 1 & \text{if } V \cong W \\ 0 & \text{otherwise}. \end{array} \right. \]

Proof. Since \(\operatorname{Hom}_G(V,W) = \operatorname{Hom}(V,W)^G\), we apply the trace formula (Corollary 7.1) to the induced representation \(\operatorname{Hom}(V,W) \cong V^* \otimes W\) and use Schur’s Lemma.

This result is fundamental to all that follows. Its name comes from the fact that a natural inner product to place on the vector space of all functions \(G \to \mathbb{C}\) is \[ \langle f_1, f_2 \rangle = \frac{1}{|G|} \sum_{g \in G} \overline{f_1(g)} f_2(g) \tag{7.2}\]

Definition 7.2 We write \(\mathscr{L}^{2}({G})\) for the inner product space of all functions \(G \to \mathbb{C}\) equipped with the above inner product. This space is often referred to as the \(L^2\)-space of functions on \(G\).

Much of the rest of this course is devoted to studying \(\mathscr{L}^{2}({G})\). Phrased in this language, the previous theorem states \[ \langle \chi_V, \chi_W \rangle = \left\{ \begin{array}{ll} 1 & \text{if } V \cong W \\ 0 & \text{otherwise}, \end{array} \right. \tag{7.3}\] for all irreducible \(V\) and \(W\). More generally, even if \(V\) and \(W\) are not irreducible, we have \[ \langle \chi_V, \chi_W \rangle = \dim \operatorname{Hom}_G(V,W). \]

Remark 7.1. Note that \(\mathscr{C}({G})\) is a natural subspace of \(\mathscr{L}^{2}({G})\). On this subspace, we can simplify the inner product just as we did with Equation 7.1: \[ \langle f_1, f_2 \rangle = \frac{1}{|G|} \sum_{i=1}^r |\operatorname{cl}_{G}({g_i})| \overline{f_1(g_i)} f_2(g_i) \tag{7.4}\] for all \(f_1, f_2 \in \mathscr{C}({G})\).

From this key insight, a bounty of additional results spring forth!

Corollary 7.2 A representation is uniquely determined by its character. In other words, if \(V\) and \(W\) are \(G\)-representations and \(\chi_V = \chi_W\), then \(V \cong W\).

Proof. If \(V\) decomposes as \(V_1^{\oplus a_1} \oplus \cdots \oplus V_r^{\oplus a_r},\) with the \(V_i\) irreducible representations of \(G\), then \(\chi_V = a_1 \chi_{V_1} + \cdots + a_r \chi_{V_r}\) (Proposition 7.3). Because the set \(\{\chi_{V_1},\dots,\chi_{V_r}\}\) is pairwise orthonormal, it is linearly independent (Exercise B.3). Hence if \(\chi_W = \chi_V\) then \[ W \cong V_1^{\oplus a_1} \oplus \cdots \oplus V_r^{\oplus a_r} \cong V. \]

Corollary 7.3 If \(V\) decomposes into irreducibles as \(V_1^{\oplus a_1} \oplus \cdots \oplus V_r^{\oplus a_r}\), then \[ \| \chi_V \|^2 = a_1^2+ \cdots + a_r^2. \] More generally, if \(W = V_1^{\oplus b_1} \oplus \cdots \oplus V_r^{\oplus b_r}\), then the inner product with \(\chi_V\) is given by \[ \langle \chi_V, \chi_W \rangle = a_1 b_1 + \cdots + a_r b_r, \] which is equal to the dimension of \(\operatorname{Hom}_G(V,W)\).

Proof. By orthonormality, we have: \[ \begin{split} \langle \chi_V, \chi_W \rangle & = \langle a_1 \chi_{V_1} + \cdots + a_r \chi_{V_r}, b_1 \chi_{V_1} + \cdots + b_r \chi_{V_r} \rangle \\ & = \sum_{i, j = 1}^r \overline{a_i} b_j \langle \chi_{V_i}, \chi_{V_j} \rangle \\ & = \sum_{i=1}^r a_i b_i, \end{split} \] where complex conjugation is ignored because the multiplicities \(a_i\) are integers.

Corollary 7.4 A representation \(V\) is irreducible if and only if \[ \| \chi_V \|^2 = 1. \]

Proof. A sum of squares is equal to \(1\) if and only if every term is \(0\) except one.

Corollary 7.5 (Computing isotypic decompositions) Let \(V\) and \(V_i\) be representations of \(G\), where \(V_i\) is irreducible. Then the multiplicity of \(V_i\) in \(V\) is given by \[ a_i = \langle \chi_{V_i}, \chi_V \rangle. \]

Proof. We know \(V = V_1^{\oplus a_1} \oplus \cdots \oplus V_r^{\oplus a_r}\), even if the multiplicities are unknown. Hence \[ \langle \chi_{V_i}, \chi_V \rangle = \sum_{j=1}^r a_j \langle \chi_{V_i}, \chi_{V_j} \rangle = a_i. \]

Corollary 7.6 The number of irreducible representations of \(G\) is less than or equal to the number of conjugacy classes of \(G\).

Proof. If \(V_1, \dots, V_r\) is a list of the irreducible representations of \(G\), then the set of corresponding characters \(\{\chi_1, \dots, \chi_r\}\) is pairwise orthonormal (and therefore linearly independent) in the space of class functions \(\mathscr{C}({G})\). Since \(\dim \mathscr{C}({G})\) is the number of conjugacy classes of \(G\), the desired inequality follows.

Recall the standard representation (Example 4.16) of \(\mathcal{S}_{n}\), given as the orthogonal complement \[ V = \left\{ \alpha_1 e_1 + \cdots + \alpha_n e_n: \alpha_1 + \cdots + \alpha_n = 0 \right\} \] in the permutation representation.

Proposition 7.5 For any \(n \geq 2\), the standard representation \(\mathcal{S}_{n} ⟳V\) is irreducible.

Proof. We begin by considering the permutation representation \(\mathcal{S}_{n} ⟳\mathbb{C}^n\). By Exercise A.6, we want to compute \[ \langle \chi_{\mathbb{C}^n}, \chi_{\mathbb{C}^n} \rangle = \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_{n}} \left|\operatorname{Fix}(\sigma)\right|^2 \] Note that \(\left|\operatorname{Fix}(\sigma)\right|^2 = \left| \operatorname{Fix}(\sigma) \times \operatorname{Fix}(\sigma) \right|\), where the rightmost set consists of pairs \((i,j)\) such that \(\sigma(i) = i\) and \(\sigma(j) = j\). We proceed by counting permutations that fix pairs: \[ \sum_{\sigma \in \mathcal{S}_{n}} \left| \operatorname{Fix}(\sigma)\right|^2 = \left| \{ (\sigma, i, j) : \sigma(i) = i, \sigma(j) = j \} \right| = \sum_{1 \leq i,j \leq n} \left| \{ \sigma: \sigma(i) = i, \sigma(j) = j \} \right|. \] There are two cases to consider, namely whether \(i\) and \(j\) agree. If \(i=j\), there are \((n-1)!\) permutations fixing \(i\), and there are \(n\) possible values which \(i\) can take on; if \(i \not = j\), there are \((n-2)!\) permutations fixing them, and there are \(n(n-1)\) such values of \(i\) and \(j\). Hence \[ \sum_{\sigma \in \mathcal{S}_{n}} \left|\operatorname{Fix}(\sigma)\right|^2 = n \cdot (n-1)! + n(n-1) \cdot (n-2)! = 2n!, \] i.e., \(\langle \chi_{\mathbb{C}^n}, \chi_{\mathbb{C}^n} \rangle = 2\). But this means that \[ \begin{split} \langle \chi_V, \chi_V \rangle & = \langle \chi_{\mathbb{C}^n} - 1, \chi_{\mathbb{C}^n} - 1 \rangle \\ & = \langle \chi_{\mathbb{C}^n}, \chi_{\mathbb{C}^n} \rangle - \langle \chi_{\mathbb{C}^n}, 1 \rangle - \langle 1, \chi_{\mathbb{C}^n} \rangle + \langle 1, 1 \rangle \\ & = 2 - 1 - 1 + 1 = 1, \end{split} \] where \(\langle \chi_{\mathbb{C}^n}, 1 \rangle = 1\) is recording the multiplicity of the trivial representation in \(\mathbb{C}^n\).

Proposition 7.6 Let \(V_i\) be an irreducible representation of \(G\). Then the multiplicity of \(V_i\) in the group algebra \(\mathbb{C}[G]\) is \(\dim V_i\).

Proof. Recall (Exercise D.6) that the action of \(G\) fixes no basis element of \(\mathbb{C}[G]\) for any \(g \not = e\), so \(\chi_{\mathbb{C}[G]}(g) = 0\). Moreover, \(\chi_{\mathbb{C}[G]}(e) = \dim \mathbb{C}[G] = |G|\), so we can compute: \[ \langle \chi_{V_i}, \chi_{\mathbb{C}[G]} \rangle = \frac{1}{|G|} \left( \chi_{V_i}(e) |G| \right) = \dim V_i. \] Thus \(\mathbb{C}[G] \cong {V_1}^{\oplus \dim V_1} \oplus \cdots \oplus {V_d}^{\oplus \dim V_d}\).

By taking dimensions, we have the following:

Theorem 7.2 If \(V_1, \dots, V_r\) are the irreducible representations of \(G\), then \[ |G| = (\dim V_1)^2 + \cdots + (\dim V_r)^2. \]

By Proposition 5.5, we have:

Corollary 7.7 Every finite abelian group \(A\) has \(|A|\) distinct irreducible representations.