Complex Element Representations

Introduction

This article introduces the representation theory of the complex algebra. The goal is to define representations precisely, classify them, and describe the structure of the representation ring.

The treatment is mathematically honest: every claim is either proved or stated as a definition. The complex algebra is assumed from the article on complex algebra, and the identification of $\mathbb{C}$ with $\mathbb{R}^2$ is used throughout. No physics is invoked.

Representations of $\mathbb{C}$

Definition

A representation of $\mathbb{C}$ is a complex vector space $V$ together with a bilinear map

$$ \rho : \mathbb{C} \times V \to V, \qquad \rho(A, v) = A \cdot v, $$

satisfying

$$ A \cdot (B \cdot v) = (AB) \cdot v, \qquad 1 \cdot v = v. $$

Equivalently, a representation is an algebra homomorphism

$$ \rho : \mathbb{C} \to \operatorname{End}(V). $$

Since $\mathbb{C}$ is a field, every representation is a complex vector space, and the action is by complex scalar multiplication.

The Regular Representation

The regular representation of $\mathbb{C}$ is $\mathbb{C}$ acting on itself by left multiplication:

$$ \rho_{\mathrm{reg}}(A) B = A B, \qquad A, B \in \mathbb{C}. $$

This is the representation $\rho_{\mathrm{reg}} : \mathbb{C} \to \operatorname{End}(\mathbb{C})$ given by $\rho_{\mathrm{reg}}(A) = A$. It is the representation of $\mathbb{C}$ on a one-dimensional complex vector space.

The Scalar Maps

For each $\lambda \in \mathbb{C}$, the scalar map $\rho_\lambda$ is defined on $\mathbb{C}$ by

$$ \rho_\lambda(A) B = \lambda A B, \qquad A, B \in \mathbb{C}. $$

Equivalently, $\rho_\lambda(A) = \lambda A$ as a complex-linear map. Since a representation must satisfy $1 \cdot B = B$, this is a representation only for $\lambda = 1$, which is the regular representation; $\lambda = 0$ gives the zero map, which is a representation only on the zero space.

Classification

The Classification Theorem

Theorem. Every finite-dimensional complex representation of $\mathbb{C}$ is isomorphic to a direct sum of copies of the regular representation:

$$ V \cong \mathbb{C}^{\oplus n}, \qquad \rho(A)(v_1, \dots, v_n) = (A v_1, \dots, A v_n). $$

Proof. The map $A \mapsto A \cdot v$ is $\mathbb{C}$-linear in $A$ for each fixed $v$, because $\rho$ is a $\mathbb{C}$-algebra homomorphism; hence $A \cdot v = A\,(1 \cdot v) = A v$ for every $v$. In particular $A \cdot v_i = A v_i$ on a basis, so the action is the scalar multiplication $A \cdot v = Av$ and $V \cong \mathbb{C}^{\oplus n}$ as a representation.

Irreducible Representations

Theorem. The regular representation is the unique irreducible complex representation of $\mathbb{C}$, up to isomorphism.

Proof. Any irreducible representation is a quotient of the regular representation, hence isomorphic to it, since $\mathbb{C}$ is a field and every non-zero $\mathbb{C}$-linear map $\mathbb{C} \to V$ is injective.

Schur's Lemma

Theorem (Schur). Every $\mathbb{C}$-linear endomorphism of an irreducible complex representation of $\mathbb{C}$ is a scalar multiple of the identity.

Proof. Let $V$ be irreducible and let $T : V \to V$ be $\mathbb{C}$-linear and commuting with the action. Then $\ker T$ and $\operatorname{im} T$ are subrepresentations. Since $V$ is irreducible, either $\ker T = 0$ and $\operatorname{im} T = V$ (so $T$ is an isomorphism), or $\ker T = V$ (so $T = 0$). In the first case, $T$ is an isomorphism, and since $V$ is one-dimensional over $\mathbb{C}$ (by the classification theorem), $T$ is multiplication by a non-zero scalar.

Corollary. The endomorphism ring of the regular representation is $\mathbb{C}$ itself.

The Representation Ring

Definition

The representation ring $R(\mathbb{C})$ is the Grothendieck ring of finite-dimensional complex representations of $\mathbb{C}$. As an abelian group, it is generated by the isomorphism class $[\rho_{\mathrm{reg}}]$, with no relations. So

$$ R(\mathbb{C}) \cong \mathbb{Z}, $$

with generator $[\rho_{\mathrm{reg}}]$.

Ring Structure

The product in $R(\mathbb{C})$ is given by the tensor product of representations:

$$ [V] \cdot [W] = [V \otimes_{\mathbb{C}} W]. $$

Since the regular representation is one-dimensional, its tensor powers are

$$ \rho_{\mathrm{reg}} \otimes \rho_{\mathrm{reg}} \cong \rho_{\mathrm{reg}}. $$

So the multiplication in $R(\mathbb{C})$ is the ordinary multiplication in $\mathbb{Z}$.

Representations and the Field Structure

The Role of the Field

The representation theory of $\mathbb{C}$ is trivial because $\mathbb{C}$ is a field. Every module over a field is free, so every representation is a direct sum of copies of the regular representation. The irreducible representations are one-dimensional over $\mathbb{C}$, and there is exactly one of them, up to isomorphism.

This is a general fact about fields: if $F$ is any field, the finite-dimensional representations of $F$ are all direct sums of copies of the regular representation, and the irreducible representations are one-dimensional. The representation ring is $\mathbb{Z}$, generated by the regular representation.

The Role of the Complex Structure

The complex structure enters only through the choice of the coefficient field. If we forget the complex structure and consider $\mathbb{C}$ as a real algebra, the representations are real vector spaces and the action is real-linear, but $\mathbb{C}$ is still a division algebra, so there is exactly one irreducible representation, of real dimension two.

So the representation theory of $\mathbb{C}$ is trivial as a complex algebra, and it remains trivial as a real algebra. The distinction is the distinction between the field $\mathbb{C}$ and the real algebra $\mathbb{C}$, and it is the reason the complex case is both the simplest and the most subtle.

The Dual Representation

Definition

The dual (or contragredient) representation of a representation $\rho$ on $V$ is the representation $\rho^*$ on the dual space $V^* = \operatorname{Hom}_{\mathbb{C}}(V, \mathbb{C})$ defined by

$$ (\rho^*(A) f)(v) = f(\rho(A) v), \qquad A \in \mathbb{C}, \; f \in V^*, \; v \in V. $$

Basic Properties

Duality is an involution. $(V^*)^* \cong V$ for finite-dimensional $V$.

Duality is exact. It preserves direct sums: $(V \oplus W)^* \cong V^* \oplus W^*$.

The dual of an irreducible is irreducible. $\rho_{\mathrm{reg}}^* \cong \rho_{\mathrm{reg}}$.

The dual of the regular representation is the regular representation. $\rho_{\mathrm{reg}}^* \cong \rho_{\mathrm{reg}}$.

The Pairing

The natural pairing

$$ \langle \cdot, \cdot \rangle : V^* \times V \to \mathbb{C}, \qquad \langle f, v \rangle = f(v), $$

satisfies

$$ \langle \rho^*(A) f, v \rangle = \langle f, \rho(A) v \rangle. $$

This is the definition of the dual representation, written as a pairing.

Tensor Products

Definition

The tensor product of two representations $V$ and $W$ is the representation on $V \otimes_{\mathbb{C}} W$ defined by

$$ A \cdot (v \otimes u) = (A \cdot v) \otimes u = v \otimes (A \cdot u). $$

The two definitions agree because $\mathbb{C}$ is commutative.

Basic Properties

Associativity. $(U \otimes V) \otimes W \cong U \otimes (V \otimes W)$.

Commutativity. $V \otimes W \cong W \otimes V$.

Distributivity. $U \otimes (V \oplus W) \cong (U \otimes V) \oplus (U \otimes W)$.

Tensor product of irreducibles. The tensor product of two copies of the regular representation is the regular representation:

$$ \rho_{\mathrm{reg}} \otimes \rho_{\mathrm{reg}} \cong \rho_{\mathrm{reg}}. $$

Homomorphisms

Definition

A homomorphism of representations $V$ and $W$ is a complex-linear map $T : V \to W$ such that

$$ T(A \cdot v) = A \cdot T(v), \qquad A \in \mathbb{C}, \; v \in V. $$

The space of all such homomorphisms is denoted $\operatorname{Hom}_{\mathbb{C}}(V, W)$.

Basic Properties

Composition. If $T \in \operatorname{Hom}_{\mathbb{C}}(V, W)$ and $S \in \operatorname{Hom}_{\mathbb{C}}(W, U)$, then $S T \in \operatorname{Hom}_{\mathbb{C}}(V, U)$.

Schur's lemma. If $V$ and $W$ are irreducible, then $\operatorname{Hom}_{\mathbb{C}}(V, W) = 0$ if $V \not\cong W$, and $\operatorname{Hom}_{\mathbb{C}}(V, V) \cong \mathbb{C}$.

Dimension count. For $V \cong \mathbb{C}^{\oplus p}$ and $W \cong \mathbb{C}^{\oplus q}$,

$$ \dim_{\mathbb{C}} \operatorname{Hom}_{\mathbb{C}}(V, W) = pq. $$

Proof. A homomorphism is a complex-linear map between finite-dimensional complex vector spaces, so it is determined by a $q \times p$ complex matrix, of complex dimension $pq$.

The Endomorphism Ring

The endomorphism ring of a representation $V$ is $\operatorname{End}_{\mathbb{C}}(V) = \operatorname{Hom}_{\mathbb{C}}(V, V)$. For $V \cong \mathbb{C}^{\oplus n}$,

$$ \operatorname{End}_{\mathbb{C}}(V) \cong M_n(\mathbb{C}), $$

the ring of $n \times n$ complex matrices. This is a simple ring, and it is the prototypical example of a central simple algebra over $\mathbb{C}$.

Comparison with the Split Complex Case

The representation theory of $\mathbb{C}$ differs from that of $\mathbb{D}$ in four essential ways.

Field versus ring. $\mathbb{C}$ is a field, so every non-zero element is invertible. $\mathbb{D}$ is not a field, so it has zero divisors. This is the source of all the differences.

Number of irreducibles. $\mathbb{C}$ has one irreducible representation, up to isomorphism. $\mathbb{D}$ has two: $\rho_+$ and $\rho_-$.

Endomorphism rings. The endomorphism ring of the regular representation of $\mathbb{C}$ is $\mathbb{C}$ itself. The endomorphism ring of the regular representation of $\mathbb{D}$ is $\mathbb{R} \oplus \mathbb{R}$, which is isomorphic to $\mathbb{D}$.

Representation rings. $R(\mathbb{C}) \cong \mathbb{Z}$, generated by the regular representation. $R(\mathbb{D}) \cong \mathbb{Z} \oplus \mathbb{Z}$, generated by $\rho_+$ and $\rho_-$. The two rings are not isomorphic: the first has no idempotents other than $0$ and $1$, while the second has a non-trivial idempotent corresponding to $\rho_+$.

Summary

A representation of $\mathbb{C}$ is a complex vector space $V$ together with a bilinear action of $\mathbb{C}$ on it, equivalently an algebra homomorphism $\mathbb{C} \to \operatorname{End}(V)$. Because $\mathbb{C}$ is a field the theory is completely determined: every module over a field is free, and every finite-dimensional representation is isomorphic to a direct sum of copies of the regular representation, $V \cong \mathbb{C}^{\oplus n}$. The regular representation is therefore the unique irreducible representation up to isomorphism, and by Schur's lemma its endomorphism ring is $\mathbb{C}$ itself.

The representation ring $R(\mathbb{C})$ is the Grothendieck ring of the finite-dimensional representations. It is generated as an abelian group by the class of the regular representation with no relations, so $R(\mathbb{C}) \cong \mathbb{Z}$. The field structure is the source of this simplicity and of every contrast with the split complex case, where $\mathbb{D}$ is a ring with zero divisors rather than a field.

The article also records the standard constructions on representations: the dual or contragredient representation on $V^*$, defined by $(\rho^*(A)f)(v) = f(\rho(A)v)$; the tensor product, whose action is well defined because $\mathbb{C}$ is commutative; and the homomorphisms, the complex-linear maps intertwining the two actions, with their composition and functorial behaviour. The final section sets out the four essential differences from the representation theory of $\mathbb{D}$.

Summary of Notation

Further Reading

• Charles C. Pinter, A Book of Abstract Algebra (Dover, 2010), for the representation theory of algebras.
• Israel Nathan Herstein, Noncommutative Rings (Mathematical Association of America, 1968), for the general theory of modules over rings.
• Serge Lang, Algebra (Springer, 2002), for the structure theory of semisimple algebras.
• John H. Conway and Derek A. Smith, On Quaternions and Octonions (A K Peters, 2003), for the classification of real algebras.
• Vladimir V. Kisil, Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of SL(2, ℝ) (Imperial College Press, 2012), for the analytic applications.