Real Element Representations

Introduction

This article introduces the representation theory of the real 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 real algebra is assumed from the article on real algebra. The order of $\mathbb{R}$ is used where positive-definiteness appears; completeness, developed in the preceding article, turns out to play no role in the representation theory. No physics is invoked.

Representations of $\mathbb{R}$

Definition

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

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

satisfying

$$ a \cdot (b \cdot v) = (ab) \cdot v, \qquad 1 \cdot v = v. $$

Equivalently, a representation is an algebra homomorphism

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

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

The Regular Representation

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

$$ \rho_{\mathrm{reg}}(a) b = a b, \qquad a, b \in \mathbb{R}. $$

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

Classification

The Classification Theorem

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

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

Proof. For fixed $v \in V$, the map $a \mapsto a \cdot v$ is $\mathbb{R}$-linear in $a$, being the composite of the $\mathbb{R}$-linear map $\rho : \mathbb{R} \to \operatorname{End}(V)$ with evaluation at $v$. It is therefore determined by its value at $1$, which is $v$: hence

$$ a \cdot v = a(1 \cdot v) = a v $$

for every $a \in \mathbb{R}$ and every $v \in V$, where the scalar $a$ on the right multiplies in $V$. So the action of the field element $a$ is already the scalar multiplication of $V$, and in any basis the matrix of $\rho(a)$ is diagonal with every entry equal to $a$. The representation is thus a direct sum of copies of the regular representation.

Irreducible Representations

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

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

Schur's Lemma

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

Proof. Let $V$ be irreducible and let $T : V \to V$ be $\mathbb{R}$-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 $\mathbb{R}$ (by the classification theorem), $T$ is multiplication by a non-zero scalar.

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

The Representation Ring

Definition

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

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

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

Ring Structure

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

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

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

Representations and the Field Structure

The Role of the Field

The representation theory of $\mathbb{R}$ is trivial because $\mathbb{R}$ 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 $\mathbb{R}$, 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 Order

The order on $\mathbb{R}$ does not enter the representation theory, because representations are linear and the order is not a linear-algebraic datum: an $\mathbb{R}$-linear map need not respect the order. The order is, however, what makes the notion of a positive-definite invariant inner product available, and the theorem below shows that in this case it imposes no restriction, since every finite-dimensional representation admits one.

A representation $V$ admits a positive-definite invariant inner product if there exists an inner product $\langle \cdot, \cdot \rangle$ on $V$ such that

$$ \langle a \cdot v, w \rangle = \langle v, a \cdot w \rangle, \qquad a \in \mathbb{R}, \; v, w \in V. $$

Theorem. Every finite-dimensional real representation of $\mathbb{R}$ admits a positive-definite invariant inner product.

Proof. The representation is a direct sum of copies of the regular representation, and on the regular representation the standard inner product is invariant: $\langle a \cdot v, w \rangle = (av)w = v(aw) = \langle v, a \cdot w \rangle$. The orthogonal direct sum of the standard inner products on the summands is then an invariant positive-definite inner product on $V$.

The Role of Completeness

Completeness does not in fact enter the representation theory, even for infinite-dimensional representations. A Banach representation of $\mathbb{R}$ is a Banach space $V$ together with a continuous algebra homomorphism $\rho : \mathbb{R} \to B(V)$, where $B(V)$ is the algebra of bounded linear operators on $V$.

Theorem. Every Banach representation of $\mathbb{R}$ is a direct sum of copies of the regular representation, in the sense that $V$ is a direct sum of closed invariant subspaces on each of which $\mathbb{R}$ acts by scalar multiplication.

Proof. Since $\rho$ is an algebra homomorphism of real algebras it is $\mathbb{R}$-linear and unital, so $\rho(a) = \rho(a \cdot 1) = a\,\rho(1) = a\,\mathrm{id}$ for every $a \in \mathbb{R}$: the field $\mathbb{R}$ acts on $V$ by scalar multiplication. Neither continuity nor completeness is used, and the conclusion holds for every representation, finite- or infinite-dimensional, Banach or not.

The Dual Representation

Definition

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

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

Basic Properties

Duality is an involution. For finite-dimensional $V$, $(V^*)^* \cong V$. Finite dimension is needed here: for an infinite-dimensional space the double dual is strictly larger than the space.

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

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

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 $V \otimes_{\mathbb{R}} W$ defined by

$$ a \cdot (v \otimes w) = (a \cdot v) \otimes w = v \otimes (a \cdot w). $$

The two definitions agree because $\mathbb{R}$ 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 real-linear map $T : V \to W$ such that

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

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

Basic Properties

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

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

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

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

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

The Endomorphism Ring

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

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

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

Comparison with the Complex and Split Complex Cases

The representation theory of $\mathbb{R}$ is the simplest of the three real algebras $\mathbb{R}$, $\mathbb{C}$ and $\mathbb{D}$ considered here, but it is the foundation on which the other two are built.

Field versus ring. $\mathbb{R}$ 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. $\mathbb{R}$ has one irreducible representation, up to isomorphism, of real dimension $1$. $\mathbb{D}$ has two, $\rho_+$ and $\rho_-$, each of real dimension $1$. $\mathbb{C}$, regarded as a real algebra, has one, of real dimension $2$.

Endomorphism rings. The endomorphism ring of the regular representation of $\mathbb{R}$ is $\mathbb{R}$ itself. The endomorphism ring of the regular representation of $\mathbb{D}$ is $\mathbb{R} \oplus \mathbb{R}$. The endomorphism ring of the regular representation of $\mathbb{C}$ as a real algebra is $\mathbb{C}$.

Representation rings. $R(\mathbb{R}) \cong \mathbb{Z}$, generated by the regular representation. $R(\mathbb{D}) \cong \mathbb{Z} \oplus \mathbb{Z}$, generated by $\rho_+$ and $\rho_-$. $R(\mathbb{C})$ as a complex algebra is $\mathbb{Z}$. As a real algebra it is $\mathbb{Z}$ as well, generated as a group by the class of the two-dimensional irreducible; since $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \oplus \mathbb{C}$, that class $x$ satisfies $x^2 = 2x$.

Summary

A representation of $\mathbb{R}$ is a real vector space $V$ together with a bilinear action of the field on it, equivalently a unital algebra homomorphism $\mathbb{R} \to \operatorname{End}_{\mathbb{R}}(V)$. The theory is the simplest of the number systems and it is completely determined: because $\mathbb{R}$ is a field every module over it is free, so every finite-dimensional representation is isomorphic to a direct sum of copies of the regular representation, and the regular representation is the unique irreducible representation up to isomorphism.

The representation ring $R(\mathbb{R})$ is generated as an abelian group by the class of the regular representation with no relations, so $R(\mathbb{R}) \cong \mathbb{Z}$. The field structure is the source of this simplicity, as it is in the complex case, and the theory of $\mathbb{R}$ is the foundation on which the complex and split complex cases are built.

The article also records the standard constructions on representations: the dual or contragredient representation on $V^*$, the tensor product with its diagonal action, and the homomorphisms, the real-linear maps intertwining the two actions, with their composition and functorial behaviour. The final section compares the theory with the complex and split complex cases.

Summary of Notation

Further Reading

• Charles C. Pinter, A Book of Abstract Algebra (Dover, 2010), for the representation theory of fields.
• 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.
• Walter Rudin, Real and Complex Analysis (McGraw-Hill, 1987), for the analytic background.