Spin Representations — Mathematics Knowledge System

Overview

Spin representations are the fundamental representations of the spin groups Spin(n), which are the simply-connected double covers of the special orthogonal groups SO(n). They do not descend to representations of SO(n) (since the kernel ℤ/2ℤ of Spin(n) → SO(n) acts non-trivially) — they are 'genuine' representations of the covering group. Spin representations are constructed via Clifford algebras and are central to Dirac's theory of the electron, the Atiyah–Singer index theorem, and the theory of spinors in differential geometry.

Intuition

Vectors in ℝⁿ transform under SO(n). Spinors transform under Spin(n): as you rotate 360°, a spinor picks up a sign change (it only returns to itself after a 720° rotation). This is not a mathematical trick — it is physically measurable via the Aharonov–Bohm effect. The spin representation makes this precise: it is the representation of Spin(n) that does not factor through SO(n).

Formal Definition

Definition

The Clifford algebra Cl(n) = Cl(ℝⁿ, Q) (where Q is the standard positive-definite form) is the associative algebra generated by e₁,…,eₙ with relations eᵢeⱼ + eⱼeᵢ = 2δᵢⱼ. The spin group Spin(n) ⊂ Cl(n)× consists of even-degree unit spinors.

\mathrm{Cl}(n) = T(\mathbb{R}^n) / \langle v^2 = -|v|^2 \cdot 1 \rangle \quad (\text{some conventions use } v^2 = +|v|^2)

Clifford algebra definition

\mathrm{Spin}(n) = \{v_1 \cdots v_{2k} \in \mathrm{Cl}(n) : v_i \in S^{n-1} \subset \mathbb{R}^n\} \subset \mathrm{Cl}(n)^\times

Spin group as unit even elements

1 \to \mathbb{Z}/2\mathbb{Z} \to \mathrm{Spin}(n) \xrightarrow{\pi} SO(n) \to 1 \quad (n \geq 2)

Double cover of SO(n) by Spin(n)

S = S^+ \oplus S^- \quad (n \text{ even}), \quad S \text{ irreducible } (n \text{ odd}); \quad \dim S = 2^{\lfloor n/2 \rfloor}

Spin representation(s): Weyl spinors for even n, Dirac spinor for odd n

Notation

Notation	Meaning
$\mathrm{Spin}(n)$	Simply-connected double cover of SO(n)
$\mathrm{Cl}(n)$	Clifford algebra of ℝⁿ with standard inner product
$S$	Spin representation (spinor space)
$S^\pm$	Positive/negative Weyl spinors (for n even)

Properties

Dimension

dim_ℂ S = 2^{⌊n/2⌋}. For n=3 (Spin(3)≅SU(2)): dim S = 2 (the standard representation). For n=4 (Spin(4)≅SU(2)×SU(2)): dim S± = 2 each.

Triality (n=8)

Spin(8) has an exceptional outer automorphism of order 3 (triality) that permutes the vector representation and the two half-spin representations S+ and S−, all of dimension 8.

Theorems

Theorem 1: Existence of Spin Representation

The Clifford algebra Cl(n) ⊗ ℂ is isomorphic to a matrix algebra (for n even: M_{2^{n/2}}(ℂ); for n odd: M_{2^{(n-1)/2}}(ℂ) ⊕ M_{2^{(n-1)/2}}(ℂ)). The spin representation(s) arise from the unique irreducible module(s) of these matrix algebras, restricted to Spin(n) ⊂ Cl(n).

Theorem 2: Spin Representations Do Not Descend to SO(n)

The element −1 ∈ Spin(n) (the non-trivial element of the kernel of Spin(n) → SO(n)) acts as −Id on S. Since −1 ≠ Id, the spin representation does not factor through SO(n).

Theorem 3: Highest Weight of Spin Representation

For SO(2n) (type Dₙ), the half-spin representations S± have highest weights ω_{n-1} and ωₙ (the two 'spinor' fundamental weights). For SO(2n+1) (type Bₙ), the spin representation S has highest weight ωₙ (the spinor fundamental weight).

Worked Examples

1
Cl(3) ⊗ ℂ ≅ M₂(ℂ) ⊕ M₂(ℂ) (as complex algebras). The even part Cl⁰(3) ⊗ ℂ ≅ M₂(ℂ) contains Spin(3). The spin representation is the unique 2-dimensional irreducible of M₂(ℂ).
$\mathrm{Cl}(3) \otimes \mathbb{C} \cong M_2(\mathbb{C}) \oplus M_2(\mathbb{C})$
2
The even part Cl⁰(3) is spanned by 1, e₁e₂, e₁e₃, e₂e₃. Setting iσₖ = eⱼeₖ (Pauli matrices), we get Cl⁰(3) ≅ ℍ (quaternions), and Spin(3) = {q ∈ ℍ : |q|=1} ≅ SU(2).
$\mathrm{Spin}(3) \cong SU(2), \quad \text{spin rep} = \text{standard 2-dim rep of } SU(2)$
3
The covering map Spin(3) → SO(3): q ∈ SU(2) acts on ℝ³ ≅ Im(ℍ) by v ↦ qvq̄. This is 2-to-1 (q and −q give the same rotation).
$\pi: SU(2) \to SO(3), \quad \pi(q)(v) = qvq^{-1}$

✓ Answer

Spin(3) ≅ SU(2) via quaternions. The spin representation is the standard 2-dimensional representation of SU(2), which does not descend to SO(3).

Practice Problems

Hardfree response

What are the dimensions of the spin representations of Spin(n) for n = 3, 4, 5, 6?

Hardfree response

Explain why spin representations are 'genuinely new' representations that cannot be seen at the SO(n) level, using the kernel of the covering map.

Quiz

Summary

Spin(n) is the simply-connected double cover of SO(n), with kernel ℤ/2ℤ.
The spin representation S is constructed from the unique irreducible module of Cl(n)⊗ℂ; dim_ℂ S = 2^{⌊n/2⌋}.
For n even, S = S+ ⊕ S− (Weyl spinors, half-spin representations).
The element −1 ∈ ker(Spin(n)→SO(n)) acts as −Id_S, so S does not descend to SO(n).
Spin(3) ≅ SU(2), Spin(4) ≅ SU(2)×SU(2), Spin(6) ≅ SU(4).

References

BookLawson, H.B. & Michelsohn, M.-L. — Spin Geometry (1989), Chapters 1–2
BookBröcker, T. & tom Dieck, T. — Representations of Compact Lie Groups (1985), Chapter 6