REAL REPRESENTATIONS OF FINITE GROUPS

STEFAN G. KARPINSKI

1. Reality

Definitions.

An element of a group is called real if it is conjugate to its inverse. A conjugacy class of a group is called real if it contains a real element. A character of a group is called real if all its values are real.

Henceforth, let G be a finite group. Notice that if one element of a conjugacy class is real then all of its elements are: let x, y, z ∈ G such that y^{−1}xy = x^{−1}, then

(z^{−1}yz)^{−1}(z^{−1}xz)(z^{−1}yz) = (z^{−1}y^{−1}z)(z^{−1}xz)(z^{−1}yz) =
z^{−1}y^{−1}xyz = z^{−1}x^{−1}z = (z^{−1}xz)^{−1}.

Proposition 1.1. The number of real conjugacy classes of G is equal to the number of real, irreducible characters of G.

Proof. Let M be the invertible matrix of the character table for G, and let \overline{M} denote the complex conjugate of M. For each irreducible character χ of G, \overline{χ} is also an irreducible character of G, so \overline{M} is related to M by permutation of rows:

P M = \overline{M}

for some permutation matrix P. Similarly, for each irreducible character χ on G and element g ∈ G, \overline{χ}(g) = χ(g^{−1}), so each column of M is the complex conjugate of its “inverse” column, and \overline{M} is related to M by permutation of columns:

MQ = \overline{M}

for some permutation matrix Q. Notice the significance of the matrices P and Q:

a character is real ⟺ it is fixed by P, and
a conjugacy class is real ⟺ it is fixed by Q.

Thus the number of real, irreducible characters of G is the trace of P and the number of real conjugacy classes of G is the trace of Q. But

Q = M^{−1}\overline{M} = M^{−1}P M, so tr Q = tr P.

With this, it is a simple application that the character table for any S_{n} has only real entries, since all elements of S_{n} are real and therefore so are all its characters.

Proposition 1.2. G has a non-trivial, real element iff |G| is even.

Proof. If G has even order then it contains a transposition and all transpositions are non-trivial and real. On the other hand, suppose that G has odd order and contains a real element x. So there exists y ∈ G such that y^{−1}xy = x^{−1}. But then

y^{−2}xy^{2} = y^{−1}(y^{−1}xy)y = y^{−1}x^{−1}y = (y^{−1}xy)^{−1} = x,

so y^{2} is in the normalizer x^{G} of x. Since |G| is odd, by Lagrange’s Theorem there exists a positive integer n such that y^{2n+1} = 1 and thus y = y^{2(n+1)} ∈ x^{G}; i.e., y^{−1}xy = x^{−1} = x, so x^{2} = 1, and since |G| is odd, x must be the identity.

Corollary 1.3. G has a non-trivial, real, irreducible character iff |G| is even.

Proof. This follows immediately from Propositions 1.1 and 1.2.

2. Realization

Definition. A character χ of G is said to be realizable over ℝ if there exists an ℝG-module with character χ.

Notice that by this definition, not all real characters need be realizable over ℝ. It is the aim of this paper to provide an “easy” method of determining whether a given character is realizable over ℝ or not.

Every ℝG-module can be naturally converted to a ℂG-module of equal dimension: let U be an ℝG-module and let {e_{k}} be a basis for it; then let V be the ℂG-module with basis {e_{k}} where G acts on the e_{k} as before.

It is harder to turn a given ℂG-module into a ℝG-module. Let V be a ℂGmodule with basis {e_{k}}. Let V_{ℝ} be the ℝG-module with basis {e_{k}, ie_{k}} where multiplication by G is defined by taking the complex number i to the formal i of the new basis elements. Explicitly, if e_{j}g = ∑_{k}(a_{jk} + ib_{jk})e_{k} in V for g ∈ G, then define multiplication as follows for V_{ℝ}:

(∗)  e_{j}g = ∑_{k}(a_{jk}e_{k} + b_{jk}(ie_{k})),  (ie_{j})g = ∑_{k}(−b_{jk}e_{k} + a_{jk}(ie_{k})).

The following proposition uses to ℝG-module V_{ℝ} to provide an important criterion for the realizability of irreducible characters over ℝ.

Proposition 2.1. If V is a ℂG-module with character χ, then the corresponding ℝG-module V_{ℝ} has character ψ = χ + \overline{χ}. Moreover, if V is irreducible and V_{ℝ} is reducible, then χ is realizable over ℝ.

Proof. Taking g ∈ G as above, χ(g) = ∑_{k}(a_{kk} +ib_{kk}) and by (∗), ψ(g) = 2 ∑_{k} a_{kk}, so ψ(g) = χ(g) + \overline{χ}(g). If V_{ℝ} is reducible, write V_{ℝ} = U + W with non-zero ℝGmodules U and W. If V is irreducible, then χ and \overline{χ} are each irreducible and U and W must be irreducible as well, having characters χ and \overline{χ} respectively.

3. Bilinear Forms

Definitions. Let β be a bilinear form on a vector space V over field 𝔽 with char ≠ 2.

β is called symmetric if β(u, v) = β(v, u) for all u, v ∈ V. β is called symplectic if β(u, v) = −β(v, u) for all u, v ∈ V. β is called positive-definite if β(v, v) > 0 for all non-zero v ∈ V.

If V is an 𝔽G-module, then β is called G-invariant if β(ug, vg) = β(u, v) for all u, v ∈ V and g ∈ G.

Proposition 3.1. Every ℝG-module has a G-invariant, symmetric, positive-definite, bilinear form.

Proof. Let V be an ℝG-module with basis {e_{k}} and let {g_{j}} enumerate G. For u, v ∈ V, take u_{jk}, v_{jk} ∈ ℝ such that ug_{j} = ∑_{k} u_{jk}e_{k}, and vg_{j} = ∑_{k} v_{jk}e_{k} and define

α(u, v) = ∑_{j,k} u_{jk}v_{jk}.

This is the desired G-invariant, symmetric, positive-definite, bilinear form on V.

Proposition 3.2. Let V be an ℝG-module with G-invariant, bilinear form β. If U is an ℝG-submodule of V, then so is the perpendicular space U^{⊥_{β}} defined as

U^{⊥_{β}} = {v ∈ V | β(u, v) = 0 for all u ∈ U} .

Proof. By elementary results on bilinear forms, U^{⊥_{β}} is a subspace of V. Moreover U^{⊥_{β}} is G-stable: if u ∈ U, v ∈ U^{⊥_{β}} and g ∈ G, then β(u, vg) = β(ug^{−1}, v) = 0.

Lemma 3.3. Let V be a vector space over ℝ with bilinear forms α and β such that α is positive definite. Moreover, let u, v ∈ V such that β(u, u) < 0 < β(v, v). Then there exists a basis {e_{k}} of V such that

α(e_{j}, e_{k}) = δ_{jk},
β(e_{j}, e_{k}) = 0 if j ≠ k,
β(e_{1}, e_{1}) < 0 < β(e_{2}, e_{2}).

Proof. Let \left\{{ \atop ẽ_{k}}\right\} be a basis for V that is orthogonal with respect to α, i.e.,

α(ẽ_{j}, ẽ_{k}) = δ_{jk}.

Let B = (b_{jk}) be the symmetric matrix such that b_{jk} = β(ẽ_{j}, ẽ_{k}). By a fundamental result on symmetric matrices, there exists a matrix Q = (q_{jk}) such that QQ^{t} = I (i.e., Q is orthogonal) and QBQ^{t} is diagonal. Let e_{j} = ∑_{k} q_{jk}ẽ_{k}. Then

α(e_{j}, e_{k}) = δ_{jk},  since QIQ^{t} = I, and
β(e_{j}, e_{k}) = 0 if j ≠ k,  since QBQ^{t} is diagonal.

Finally, suppose β(e_{k}, e_{k}) ≥ 0 for all k. Then β(u, u) ≥ 0 for all u ∈ V, which contradicts the hypotheses. Wlog, β(e_{1}, e_{1}) < 0 and similarly, 0 < β(e_{2}, e_{2}).

Proposition 3.4. Let V be an ℝG-module with G-invariant, symmetric, bilinear form β. If there exist u, v ∈ V such that β(u, u) < 0 < β(v, v), then V is reducible.

Proof. Let α be the G-invariant, symmetric, positive-definite, bilinear form on V as provided by Proposition 3.1, and let {e_{k}} be the basis for V as provided by Lemma 3.3. Let λ = β(e_{2}, e_{2})^{−1} ∈ ℝ and for u, v ∈ V, define

γ(u, v) = α(u, v) − λβ(u, v).

Because α and β are G-invariant, symmetric, bilinear forms on V, so is γ. Moreover, γ is a non-zero, degenerate bilinear form on V: if v ∈ V and v = ∑_{k} v_{k}e_{k} where v_{k} ∈ ℝ, then

γ(e_{1}, e_{1}) = α(e_{1}, e_{1}) − λβ(e_{1}, e_{1}) > 1,

and

γ(v, e_{2}) = v_{1}γ(e_{2}, e_{2}) = v_{2} (α(e_{2}, e_{2}) − λβ(e_{2}, e_{2})) = 0.

Thus, V^{⊥_{γ}} is a non-trivial, proper, ℝG-submodule, so V is reducible.

Theorem 3.5. An irreducible character χ of G can be realized over ℝ iff there exists a ℂG-module with character χ and a non-zero, G-invariant, symmetric, bilinear form.

Proof. If χ can be realized over ℝ, then there exists an ℝG-module U with character χ. Let α be the G-invariant, symmetric, positive-definite, bilinear form on U as provided by Proposition 3.1. Let {e_{k}} be a basis for U, and let V be the ℂGmodule with the same basis (as constructed in section 2). For u, v ∈ V such that v = ∑_{j} u_{j}e_{j} and v = ∑_{k} v_{k}e_{k} where u_{k}, v_{k} ∈ ℂ, define

β(u, v) = ∑_{j,k} u_{j}v_{k}α(e_{j}, e_{k}).

This is a non-zero, G-invariant, symmetric, bilinear form on V.

Conversely, if V is a ℂG-module with character χ and non-zero, G-invariant, symmetric, bilinear form β, then there exist u, v ∈ V such that β(u, v) ≠ 0. Observe that

β(u + v, u + v) = β(u, u) + β(v, v) + 2β(u, v),

so at least one choice of w = u, v, u + v has β(w, w) ≠ 0. Let {e_{k}} be a basis for V with e_{1} = wβ(w, w)^{−1/2}. Then {e_{k}, ie_{k}} is a basis for the ℝG-module V_{ℝ} as seen in section 2. Let σ be the natural bijection from V_{ℝ} to V: let σ: V_{ℝ} → V, taking

∑_{k}(a_{k}e_{k} + b_{k}(ie_{k})) ⟼ ∑_{k}(a_{k} + ib_{k})e_{k}

For u, v ∈ V_{ℝ} define

γ(u, v) = Re[β(σu, σv)].

Notice that γ is G-invariant, symmetric and bilinear because β is and because σ is a linear G-map. Observe that

γ(e_{1}, e_{1}) = β(e_{1}, e_{1}) = 1,  γ(ie_{1}, ie_{1}) = i^{2}γ(e_{1}, e_{1}) = −1,

so we may apply Proposition 3.4 to see that V_{ℝ} is reducible and by Proposition 2.1, χ is realizable over ℝ.

4. The Indicator Function

Definition. For χ an irreducible character of G, define the indicator ι on χ:

ιχ = ⟨χ_{S} − χ_{A}, 1_{G}⟩ = ⟨χ_{S}, 1_{G}⟩ − ⟨χ_{A}, 1_{G}⟩ .

Notice that

⟨χ_{S}, 1_{G}⟩ + ⟨χ_{A}, 1_{G}⟩ = ⟨χ_{S} + χ_{A}, 1_{G}⟩

= ⟨χ^{2}, 1_{G}⟩ = ⟨χ, \overline{χ}⟩ = \begin{cases}0, & if χ is not real, \\ 1, & if χ is real. \end{cases}

So if χ is not real then ιχ = ⟨χ_{S}, 1_{G}⟩ = ⟨χ_{A}, 1_{G}⟩ = 0, and if χ is real then either ⟨χ_{S}, 1_{G}⟩ = 1, or ⟨χ_{A}, 1_{G}⟩ = 1 but not both. Thus,

ιχ = 0, ±1 and ιχ ≠ 0 ⇔ χ is real.

Lemma 4.1. If V and W are ℂG-modules, with ℂG-homomorphism π: V → W between them, then there is a ℂG-submodule, U ⊆ V such that V = U ⊕ Ker π and U ≅ Im π.

Theorem 4.2. Let V be an irreducible ℂG-module with character χ. (1) There exists a non-zero G-invariant, bilinear form β on V iff ιχ ≠ 0.

(2) β is symmetric iff ιχ = 1, and β is symplectic iff ιχ = −1.

Proof. (1) Suppose β is a non-zero, G-invariant, bilinear form on V. Let {e_{k}} be a basis for V. Then {e_{j} ⊗ e_{k}} forms a basis for V ⊗ V. Define π: V ⊗ V → ℂ by

π(e_{j} ⊗ e_{k}) = β(e_{j}, e_{k}),

extending linearly to all of V ⊗ V. This is a non-zero ℂG-homomorphism because β is a non-zero G-invariant bilinear form. By Lemma 4.1, V ⊗ V has a trivial ℂG-submodule, so ⟨χ^{2}, 1_{G}⟩ = 1 and thus ιχ ≠ 0.

Conversely, suppose that ιχ ≠ 0. Then ⟨χ^{2}, 1_{G}⟩ = 1 and V ⊗ V has a trivial ℂG-submodule, U ≅ ℂ. Then the natural projection π: V ⊗ V → U is a non-zero ℂG-homomorphism, and β(u, v) = π(u⊗v) is a non-zero, G-invariant, bilinear form on V, since π is a non-zero ℂG-homomorphism.

(2) Since V ⊗ V = Sym(V ⊗ V) ⊕ Alt(V ⊗ V) we may write

π = π_{S} + π_{A}

where π_{S}: Sym(V ⊗ V) → ℂ, and π_{A}: Alt(V ⊗ V) → ℂ, given explicitly by

π_{S}(u ⊗ v) = \frac{1}{2}π(u ⊗ v + v ⊗ u),
π_{A}(u ⊗ v) = \frac{1}{2}π(u ⊗ v − v ⊗ u).

If ιχ = 1, then the trivial ℂG-submodule U lies entirely inside Sym(V ⊗ V), so π_{A} = 0, and β(v, u) = π_{S}(v ⊗ u) = π_{S}(u ⊗ v) = β(u, v). If ιχ = −1, then the trivial ℂG-submodule U lies entirely inside Alt(V ⊗ V), so π_{S} = 0, and β(v, u) = π_{A}(v ⊗ u) = −π_{A}(u ⊗ v) = −β(u, v).

Now it is simply a matter of pulling together the various pieces into the final result due to Frobenius and Schur, often called the “Count of Involutions,” after its last part. This result not only turns the question of realizability over ℝ into a simple matter of computing the indicator function, but also it provides a deep relationship between the internal structure of the and the values of the indicator on the irreducible characters of the group.

Corollary 4.3 (The Frobenius-Schur Count of Involutions). For each irreducible character χ of G,

(1) ιχ = \begin{cases}0, & if χ is not real, \\ 1, & if χ is realizable over ℝ, \\ −1, & if χ is not realizable over ℝ. \end{cases}

Furthermore, for all g ∈ G,

∑_{χ}(ιχ)χ(x) = |{y ∈ G | y^{2} = x}| , (2)

and in particular

∑_{χ}(ιχ)χ(1) = ∑_{χ}(ιχ)(dim χ) = 1 + t (3)

where t is the number of involutions in G.

Proof. (1) follows immediately from Theorems 3.5, and 4.2.

(2) Let ϑ: G → ℂ be defined by

ϑ(x) = |{y ∈ G | y^{2} = x}| .

Note that ϑ is a class function on G, and therefore is a linear combination of the irreducible characters of G, so we may sensibly take its inner product with χ:

⟨ϑ, χ⟩ = \frac{1}{|G|} ∑_{g∈G} ϑ(g)χ(g) = \frac{1}{|G|} ∑_{g∈G} χ(g^{2}) = ⟨χ_{S} − χ_{A}, 1_{G}⟩ = ιχ.

Writing ϑ as a sum of its constituents yields ϑ = ∑(ιχ)χ as desired.

(3) The count of involutions itself is simply the special case of (2) for g = 1.

References

1. Gordon James and Martin Liebeck, Representations and characters of groups, Cambridge Mathematical Textbooks, ch. 22 Real Representations, pp. 260–278, Cambridge University Press, Cambridge, UK, 1993.