Representation Theory of the Square: D₄, Subgroups, and Decomposition

1. Intuitive Overview: Symmetry as Linear Action

Symmetries in mathematics are transformations that leave an object the same aka indistinguishable from the pre-transformation state. For example take a square, rotate it 90 degrees, and it looks the same. Or draw a line through the middle and reflect it across the line and it looks the same. These rotations and reflections are what we call symmetries. Representation theory translates these symmetries into concrete linear transformations of a vector space: typically in form of matrices acting on coordinates. This translation into linear transformations is the mathematical formalization of the intuitive concept of symmetry.

2. Basic Definitions

Ahead of the discussion it is important to understand the following mathematical concepts.

2.1 Group

A group G is a set of elements g with a binary operation ⋅. These elements can be numbers, coordinates or whatever, and the operation can be addition, multiplication or whatever (binary just means it takes 2 arguments and puts one value out). These elements and operation must satisfy the following axioms:

• Closure: for all g₁, g₂ ∈ G, g₁ ⋅ g₂ ∈ G
• Identity: there exists an element e ∈ G such that for all g ∈ G, g ⋅ e = e ⋅ g = g
• Inverses: for all g ∈ G, there exists an element g⁻¹ ∈ G such that g ⋅ g⁻¹ = g⁻¹ ⋅ g = e
• Associativity: for all g₁, g₂, g₃ ∈ G, (g₁ ⋅ g₂) ⋅ g₃ = g₁ ⋅ (g₂ ⋅ g₃)

• Integers under addition: G = {..., -2, -1, 0, 1, 2, ...}, ⋅ = +
• Nonzero reals under multiplication: G = {x ∈ ℝ | x ≠ 0}, ⋅ = ⋅
• Symmetries of a polygon under composition: G = {rotations and reflections}, ⋅ = composition

2.2 Field

A field F is a set of elements f called scalars with two binary operations + and ⋅ satisfying the following rules:

• Closure: for all f₁, f₂ ∈ F, f₁ + f₂ ∈ F and f₁ ⋅ f₂ ∈ F
• Associativity: for all f₁, f₂, f₃ ∈ F, (f₁ + f₂) + f₃ = f₁ + (f₂ + f₃) and (f₁ ⋅ f₂) ⋅ f₃ = f₁ ⋅ (f₂ ⋅ f₃)
• Commutativity: for all f₁, f₂ ∈ F, f₁ + f₂ = f₂ + f₁ and f₁ ⋅ f₂ = f₂ ⋅ f₁
• Distributivity: for all f₁, f₂, f₃ ∈ F, f₁ ⋅ (f₂ + f₃) = (f₁ ⋅ f₂) + (f₁ ⋅ f₃)
• Identity: there exists an element 0 ∈ F such that for all f ∈ F, f + 0 = f and 1 ∈ F such that for all f ∈ F, f ⋅ 1 = f
• Additive inverses: for all f ∈ F, there exists an element −f ∈ F such that f + (−f) = 0
• Multiplicative inverses: for all f ∈ F with f ≠ 0, there exists an element f⁻¹ ∈ F such that f ⋅ f⁻¹ = 1

• Real numbers: F = ℝ, + = +, ⋅ = ⋅
• Complex numbers: F = ℂ, + = +, ⋅ = ⋅
• Rational numbers: F = ℚ, + = +, ⋅ = ⋅

2.3 Vector Space

A vector space V over a field F is a set of elements called vectors, equipped with a vector addition operation + and a scalar multiplication operation ⋅ (multiplying a vector by a scalar from F), satisfying the axioms below. The standard concrete model is Fⁿ: ordered tuples (f₁, f₂, …, fₙ) of field elements, where addition and scalar multiplication act component-wise. For example, (1, 2, 3) is a vector in ℝ³. Note that (2, 1, 3) is a different vector since the order matters. Think of a vector as such a tuple for intuition, but the abstract definition is more general (for instance, the set of all polynomials of degree at most n also forms a vector space). In the tuple model with v₁ = (f₁, f₂, f₃, ...) and v₂ = (g₁, g₂, g₃, ...): Vector addition: v₁ + v₂ = (f₁ + g₁, f₂ + g₂, f₃ + g₃, ...) field (aka scalar) multiplication: For a scalar c ∈ F, c ⋅ v₁ = (c ⋅ f₁, c ⋅ f₂, c ⋅ f₃, ...) The vector space is defined to have the following properties:

• Closure under + and scalar ⋅: for all v₁, v₂ ∈ V, v₁ + v₂ ∈ V, and for all c ∈ F and v ∈ V, c ⋅ v ∈ V
• Commutativity of +: for all v₁, v₂ ∈ V, v₁ + v₂ = v₂ + v₁
• Associativity of +: for all v₁, v₂, v₃ ∈ V, (v₁ + v₂) + v₃ = v₁ + (v₂ + v₃)
• Compatibility of scalar multiplication: for scalars f₁, f₂ ∈ F and vector v ∈ V, f₁ ⋅ (f₂ ⋅ v) = (f₁ ⋅ f₂) ⋅ v
• Additive distributivity: for any scalar f ∈ F and vectors v₁, v₂ ∈ V, f ⋅ (v₁ + v₂) = (f ⋅ v₁) + (f ⋅ v₂)
• Scalar distributivity: for any scalars f₁, f₂ ∈ F and vector v ∈ V, (f₁ + f₂) ⋅ v = (f₁ ⋅ v) + (f₂ ⋅ v)
• Additive identity: there exists an element 0 ∈ V such that for all v ∈ V, v + 0 = v
• Scalar identity: the element 1 ∈ F (from the field) satisfies 1 ⋅ v = v for all v ∈ V
• Additive inverses: for all v ∈ V, there exists an element −v ∈ V such that v + (−v) = 0

• ℝ over ℝ: the real line as a 1D vector space over itself
• ℚ over ℚ: the rationals as a 1D vector space over themselves
• ℝ² over ℝ: vectors (x, y) with real entries
• ℂ² over ℂ: vectors (z₁, z₂) with complex entries
• ℝ³ over ℝ: vectors (x, y, z) with real entries

2.4 GL(V)

The general linear group GL(V) is the group of all invertible linear maps V → V (equivalently, all invertible matrices once a basis is chosen). A linear map is a function that preserves the vector addition and scalar multiplication operations. Let V and W be vector spaces over the same field F. A function T: V → W is a linear map if it satisfies:

• For all v ∈ V and for all f ∈ F, T(f ⋅ v) = f ⋅ T(v)
• For all v₁, v₂ ∈ V, T(v₁ + v₂) = T(v₁) + T(v₂)

2.5 Homomorphism

A homomorphism is a function mapping from a group G with group operation ⋅G to a group H with group operation ⋅H (can be addition, multiplication, etc.): φ: G → H The homomorphism φ has the property that for all g₁, g₂ ∈ G, φ(g₁ ⋅G g₂) = φ(g₁) ⋅H φ(g₂). If a homomorphism is also a bijection, then it is called an isomorphism (google for definition).

From now on we will not write out the group operation explicitly (like ⋅G or ⋅H). When we write products like g₁g₂ or φ(g₁)φ(g₂), the group operation (multiplication, addition, etc.) is implicit, depending on the group involved. For addition (such as in abelian groups or vector spaces), we will write v + w without any subscript, as usual.

Examples:

• The identity homomorphism: φ(g) = g (from any group to itself)
• The determinant homomorphism: det: GL(V) → F*, where det(AB) = det(A)det(B)
• The sign homomorphism: sgn: Sn → {+1, −1}, mapping each permutation to its sign (even → +1, odd → −1)

Schur's Lemma: Let G be a group (not necessarily finite), and let V and W be irreducible representations (see definition below) of G over a field F. If φ: V → W is a linear map that commutes with the group action (i.e., φ(g·v) = g·φ(v) for all g ∈ G, v ∈ V), then either:

• φ = 0 (the zero map), or
• φ is an isomorphism.

Furthermore, if V = W is finite-dimensional and F is algebraically closed, then every linear map φ: V → V that commutes with the group action is a scalar multiple of the identity: φ = λ·id_V for some λ ∈ F.

2.6 Representation

A (linear) representation of a group G over a field F is a homomorphism ρ: G → GL(V). It assigns to each g ∈ G an invertible linear operator ρ(g) on V such that ρ(g₁g₂) = ρ(g₁)ρ(g₂), ρ(e) = I, and ρ(g⁻¹) = ρ(g)⁻¹. The pair (V, ρ) is the representation space.

2.7 Subrepresentation and Irreducibility

A subspace W ⊆ V is invariant (a subrepresentation) if ρ(g)W ⊆ W for all g ∈ G. A representation is irreducible if it has no nontrivial invariant subspaces (only {0} and V). Decomposing a representation into irreducible summands is a central aim of the theory.

2.8 Direct Sum

If (V₁, ρ₁) and (V₂, ρ₂) are representations, their direct sum is V₁ ⊕ V₂ with action ρ(g)(v₁, v₂) = (ρ₁(g)v₁, ρ₂(g)v₂). Conceptually, this puts two independent representations side by side.

3. The Symmetry Group of the Square: D₄

To see how representations work let's look at the symmetries of a square. The square is centered at the origin in the plane. Its symmetries form the dihedral group D₄ (This is just a fancy name don't worry about it): The order of the group is the number of elements in the group, which is 8, because there are four rotations and four reflections. Throughout this section we use the left-action convention: matrices act on column vectors from the left, and in a product g₁g₂ the element g₂ acts first, then g₁. That is, ρ(g₁g₂)v = ρ(g₁)(ρ(g₂)v).

3.1 Elements and Relations

• Rotations: e (0°), r (90°), r² (180°), r³ (270°)
• Reflections: across x-axis (s), y-axis (sr²), and the two diagonals (sr, sr³)

You can understand these the following: e means nothing is changed - the identity operation - , r is a rotation by 90 degrees, r² is r ⋅ r two rotations by 90 degrees chained right after each other, which results in a rotation by 180 degrees, r³ is a 3 rotations by 90° chained together resulting in a rotation by 270 degrees, s is a reflection across the x-axis, sr² is two rotations by 90 degrees followed by a reflection across the x-axis which is the same as a reflection across the y-axis, and similarly sr is a reflection across the line y = −x, sr³ is a reflection across the line y = x

Presentation and core relations: r⁴ = e, s² = e, srs = r⁻¹.

3.2 Standard 2×2 Real Matrix Representation

The interesting thing now is that we can CHOOSE how to represent this square, the choice of the field already matters. We can choose to represent the square as a vector space over the real numbers ℝ or the complex numbers ℂ. Over ℝ, the square lives in the plane ℝ² and the group acts on every point (x, y) of the ambient plane, not just the four corner vertices. (Side note: one could also build a 4D permutation representation that shuffles the four vertices; that is a different, reducible representation.) Over ℂ, we can also work in ℂ², where the same 2×2 matrices act on complex column vectors. The key difference is that over ℂ, matrices of finite order (like our rotation and reflection matrices) are always diagonalizable, because ℂ is algebraically closed and their minimal polynomials have distinct roots. This means representations that are irreducible over ℝ can sometimes split into smaller pieces over ℂ. (Side note: not every complex matrix is diagonalizable in general, e.g. Jordan blocks, but matrices arising from finite group representations over ℂ always are.) We will see this concretely in section 4. For this example we will choose ℝ. Then the representation of the square is a homomorphism ρ: D4 → GL(ℝ²). The group D4 acts on the square by rotating and reflecting it. The homomorphism ρ assigns to each element of D4 a linear transformation of ℝ² which takes the form of a 2x2 matrix acting on column vectors (x, y) in ℝ². We define our two generators explicitly to fix conventions:

• r = counterclockwise rotation by 90°: ρ(r) = [[0, -1], [1, 0]], mapping (x, y) → (−y, x)
• s = reflection across the x-axis (the line y = 0): ρ(s) = [[1, 0], [0, -1]], mapping (x, y) → (x, −y)

All other group elements are products of these two. The full set of eight matrices is:

• ρ(e) = [[1, 0], [0, 1]]
• ρ(r) = [[0, -1], [1, 0]]
• ρ(r²) = [[-1, 0], [0, -1]]
• ρ(r³) = [[0, 1], [-1, 0]]
• ρ(s) = [[1, 0], [0, -1]] (reflect across x-axis)
• ρ(sr²) = [[-1, 0], [0, 1]] (reflect across y-axis)
• ρ(sr) = [[0, -1], [-1, 0]] (reflect across line y = −x)
• ρ(sr³) = [[0, 1], [1, 0]] (reflect across line y = x)

These eight matrices are essentially fancy expressions for "rotating and reflecting the square in ℝ²" like the concept of the group D₄ describes.

3.3 Alternative Rotation Models

• Continuous rotation in the plane (group SO(2)): R(θ) = [[cosθ, −sinθ], [sinθ, cosθ]].
• Complex form for rotations only: act on ℂ by multiplication z ↦ eiθ z. For discrete θ = k·(π/2), this realizes the rotation subgroup C₄ as complex phases. Note that this only covers rotations. Geometrically, reflection across the real axis corresponds to complex conjugation z → z̄, but conjugation is not complex-linear, it is antilinear. A map T: ℂ → ℂ is called antilinear (or conjugate-linear) if T(f₁z + f₂w) = f̄₁ · T(z) + f̄₂ · T(w) for all f₁, f₂ ∈ ℂ. Conjugation satisfies this: conj(f₁z) = f̄₁ · conj(z). Since antilinear maps are not linear, conjugation does not define a representation on ℂ as a 1D complex vector space. To represent the full D₄ including reflections, we need the 2D representation over ℝ (or equivalently, 2D over ℂ) as in section 3.2.
• Higher-dimensional embeddings (e.g., 4×4 blocks) can represent combined or repeated actions; these are equivalent ways to represent the same symmetry concept in different coordinates.

4. Irreducibility of the Standard 2D Representation

Over ℝ², the standard 2D representation of D₄ is irreducible. No nontrivial 1D subspace (line through the origin) is invariant under all eight matrices: a 90° rotation sends any chosen line to a different line, destroying invariance. (Side note: the origin {0} is always invariant under every linear map, since ρ(g)·0 = 0. This is why the definition of irreducibility explicitly excludes {0} and V as trivial invariant subspaces.)

Over ℂ, the same 2D representation becomes reducible for the rotation subgroup: rotation by 90° has eigenvalues i and −i with eigenvectors proportional to (1, −i) and (1, i) respectively, so the complexified rotation action splits into two 1D characters. Reflections swap these two complex lines, which is why the full D₄-action remains irreducible over both ℝ and ℂ: the reflections "mix" the complex eigenspaces and prevent the 2D representation from splitting.

5. Restricting to Subgroups: C₄ (Rotations) and C₂ (One Reflection)

5.1 Rotation Subgroup C₄ = ⟨r⟩

Restricting the standard 2D representation to C₄ (elements e, r, r², r³):

• Over ℝ, the 2D representation remains irreducible (90° rotation has no real eigenvectors).
• Over ℂ, it decomposes into two 1D irreducible characters: χ±(r) = e±iπ/2 = ±i.

5.2 Reflection Subgroup C₂ = ⟨s⟩

Restricting to the subgroup generated by a single reflection (e.g., s = mirror across the x-axis): ρ(s) = [[1, 0], [0, −1]].

• The x-axis is the (+1)-eigenspace: span{(1, 0)}.
• The y-axis is the (−1)-eigenspace: span{(0, 1)}.

Therefore, as a representation of C₂, ℝ² splits as a direct sum of two 1D irreducibles (trivial and sign). This is a clean example of reducibility after restriction to a subgroup.

5.3 Subrepresentation vs. Restriction

A subrepresentation is an invariant subspace for the same group action. The axes are not invariant under all of D₄ (a 90° rotation moves the x-axis to the y-axis), so they are not subrepresentations of D₄. They become invariant only after restricting the acting group to a reflection subgroup (C₂).

6. Character Table of D₄

Conjugacy classes of D₄ and their sizes:

• {e} (size 1)
• {r²} (size 1)
• {r, r³} (size 2)
• {s, sr²} (size 2) — reflections across axes
• {sr, sr³} (size 2) — reflections across diagonals

Irreducible representations: four 1D (A₁, A₂, B₁, B₂) and one 2D (E).

The character of a representation is the trace of its matrices. For example, in the 2D standard representation E: χE(e)=2, χE(r²)=−2, χE(r)=χE(r³)=0, and all reflections have trace 0. Orthogonality of characters confirms completeness and enables decomposition of any representation into irreducibles. (Side note: these orthogonality relations rely on Maschke's theorem, which guarantees that every representation decomposes as a direct sum of irreducibles. This holds for finite groups over any field whose characteristic does not divide |G|. In our case |D₄| = 8 and we work over ℝ or ℂ (characteristic 0), so the theory applies without issues.)

As a consistency check, the sum of squares of the dimensions of the irreps equals the group order: 1² + 1² + 1² + 1² + 2² = 8 = |D₄|. This is a consequence of the decomposition of the regular representation: the regular representation has dimension |G| and decomposes as a direct sum of irreps, where each irrep V_i appears with multiplicity equal to its dimension, giving |G| = Σ(dim V_i)² over all irreducible representations.

7. Worked Examples and Comparisons

7.1 Standard 2D Rep Is Irreducible Over ℝ

No line through the origin is invariant under all of D₄. Rotations by 90° map each candidate line to a different one, preventing a nontrivial invariant subspace.

7.2 Restricting to Rotations C₄

Over ℝ the 2D action remains irreducible; over ℂ it splits into two 1D characters with eigenvalues i and −i. This corresponds to left- and right-circular "modes" (complex phases e±iπ/2).

7.3 Restricting to a Single Reflection C₂

For s = [[1,0],[0,−1]], the x-axis (+1-eigenspace) and y-axis (−1-eigenspace) are invariant. Thus, as a C₂-representation, ℝ² decomposes as a direct sum of two 1D irreducibles (trivial ⊕ sign).

8. Summary and Takeaways

• Representation theory turns symmetries into matrices: ρ: G → GL(V).
• D₄ (square symmetries) has a faithful 2D real representation via explicit 2×2 matrices for rotations and reflections.
• Over ℝ, this 2D D₄-representation is irreducible; over ℂ its rotation part splits into two 1D characters.
• Restriction to subgroups changes reducibility: C₄ (rotations) keeps irreducibility over ℝ but splits over ℂ; C₂ (a single reflection) splits over ℝ into (+1)- and (−1)-eigenspaces.
• The character table of D₄ encapsulates all irreducible representations and enables systematic decomposition.

Appendix: Notational Cheatsheet

• GL(V): invertible linear maps V → V (matrices in a fixed basis).
• ρ: representation map from group elements to matrices.
• χ(g) = tr(ρ(g)): character (trace) of g.
• ⊕: direct sum of representations (block-diagonal action).
• Cn: cyclic group of order n (rotations by multiples of 2π/n).
• Dn: dihedral group of order 2n (rotations and n reflections of a regular n-gon).