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 reflectionsare 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)
• 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
• Inverses: for all f ∈ F, there exists an element f⁻¹ ∈ F such that f + f⁻¹ = 0 and f ⋅ f⁻¹ = 1

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

2.3 Vector Space

A vector v is a tuple of of elements of a field (f₁ , f₂ , f₃ ...). A tuple is essentially a list, of stuff where the order matters. For example (1, 2, 3) is a vector aka tuple of the field of real numbers. (2, 1, 3) is not the same vector! The Vector lives in the so called vector space V over a field F with a vector addition operation + and a field multiplication operation ⋅ aka scalar multiplication. Let v₁ = (f₁, f₂, f₃, ...) and v₂ = (g₁, g₂, g₃, ...) be two vectors in V. Vector addition: v₁ + v₂ = (f₁ + g₁, f₂ + g₂, f₃ + g₃, ...) field 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 ⋅: for all v₁, v₂ ∈ V, v₁ + v₂ ∈ V and v₁ ⋅ v₂ ∈ V
• Associativity of + and scalar multiplication: for all v₁, v₂, v₃ ∈ V and scalars f₁, f₂, f₃ ∈ F, (v₁ + v₂) + v₃ = v₁ + (v₂ + v₃) and f₁ ⋅ (f₂ ⋅ f₃) = (f₁ ⋅ f₂ ⋅ f₃)
• 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)
• Identity aka neutral element for + and ⋅: there exists an element 0 ∈ V such that for all v ∈ V, v + 0 = v and 1 ∈ V such that for all v ∈ V, v ⋅ 1 = v
• Inverses for + and ⋅: for all v ∈ V, there exists an element v⁻¹ ∈ V such that v + v⁻¹ = 0 and v ⋅ v⁻¹ = 1

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)

2.4 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
• The inversion homomorphism: φ(g) = g⁻¹
• The conjugation homomorphism: φ(g) = g⁻¹ag

Schur's Lemma: Let G be a group, and let V and W be irreducible representations (see definition below) of G over a field F. If φ: V → W is a G-homomorphism (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 and F is algebraically closed, then φ is a scalar multiple of the identity: φ = λ·id_V for some λ ∈ F.

2.5 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.

2.6 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.

2.7 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 lets 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 illustrated in the figure below.

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° chaned 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 coice 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. For real numbers the square corner points are represented as a two dimensional vector (x, y), the so called ℝ² vector space. For complex numbers the square corner points are represented as a one dimensional complex vector (x + yi) in ℂ. 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:

• 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: act on ℂ by multiplication z ↦ eiθ z. For discrete θ = k·(π/2), this realizes the rotation subgroup C₄ as complex phases. And conjugation: z → z̄, x+yi → x-yi is the same as reflecting it over the real (x-)axis.
• 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.

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), 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 ℝ but "mixes" the complex lines when reflections are included.

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.

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).