Spinors: tables

P. Deligne

1. Complex story.

The Dynkin diagram of the group Spin(n) is:

where r is the rank. The spin or semi-spin representations are the fundamental representations corresponding to the vertex, or vertices, at the right of the diagram. If one restricts the spin representation of Spin(2r + 1) to Spin(2r), one obtains the sum of the two semispin representations of Spin(2r). Each of the two semi-spin representations of Spin(2r) restricts as the spin representation of Spin(2r − 1).

The Dynkin diagram makes clear the low rank exceptional isomorphisms. We list in the next table n = 2, 3, 4, 5 or 6, the Dynkin diagram, a group G to which Spin(n) is isomorphic, the description of spin or semi-spin representations as representations of G, and the description of the defining representation of SO(n) = Spin(n)/(ℤ/(2)) as representation of G.

n  diagram  G  (semi)spin  defining orthogonal
2  none  𝔾_{m}  characters z, z^{−1}  z^{2} ⊕ z^{−2}
3  •  SL(2)  defining V  adjoint = Sym^{2}(V)
4  ^{•} •  SL(2) × SL(2) defining V_{1}, V_{2}  V_{1} ⊗ V_{2}
5  Sp(4)  defining V  ∧^{2} V
6  SL(4)  defining V, and V^{∗}  ∧^{2} V

The symmetric bilinear form on the defining orthogonal representation is given by

n = 3: Killing form Tr(adx ady), or ⟨v^{2}, w^{2}⟩ = ψ(v, w)^{2}, for ψ symplectic (= volume) form of V.

n = 4: ⟨v_{1} ⊗ v_{2}, w_{1} ⊗ w_{2}⟩ = ψ(v_{1}, w_{1})ψ_{2}(v_{2}, w_{2}).

n = 5: ⟨v_{1}∧v_{2}, w_{1}∧w_{2}⟩ = ψ(v_{1}, w_{1})ψ(v_{2}, w_{2}) − ψ(v_{1}, w_{2})ψ(v_{2}, w_{1}).

n = 6: ⟨v_{1}∧v_{2}, w_{1}∧w_{2}⟩ = v_{1}∧v_{2}∧w_{1}∧w_{2} ∈ ∧^{4} V = ℂ.

For n odd, the spin representation S of Spin(n) is orhtogonal or symplectic. There is a morphism of representations from S ⊗ S to the defining representation V; it is symmetric or antisymmetric.

For n even, the semi-spin representations S^{+} and S^{−} are orthogonal, symplectic, or dual of each other. There is either a morphism of representations S^{+} ⊗ S^{−} → V, or morphisms S^{+} ⊗ S^{+} → V and S^{−} ⊗ S^{−} → V, symmetric or antisymmetric. Which is the case depends only on n mod 8. The next table gives n mod 8, the kind of bilinear form on spinors, and the kind of bilinear forms (spinors) ⊗ (spinors) → V.

n mod 8  forms on spinors  symmetry of spinors, spinors → V
1  orthogonal  symmetric
2  S^{+} dual to S^{−}  symmetric (on S^{+}, and on S^{−})
3  symplectic  symmetric
4  S^{+} and S^{−} sympletic  S^{+} ⊗ S^{−} → V
5  symplectic  antisymmetric
6  S^{+} dual to S^{−}  antisymmetric (on S^{+}, and on S^{−})
7  orthogonal  antisymmetric
8  S^{+} and S^{−} orthogonal  S^{+} ⊗ S^{−} → V

2. Real story.

An irreducible representation (over ℝ) of Spin(p, q) is said to be spinorial if, after extensioin of scalars to ℂ, it becomes a sum of spinorial or semi-spinorial representations.

If p−q is odd (equivalent to p+q odd), there is a unique spinorial irreducible representation. It is either real or quaternionic. if p+q is even, either there is only one such representation, in which case it is complex, giving over ℂ the sum of the two semi-spinorial representations, or there are two of them, both real or both quaternionic. If real, they remain irreducible over ℂ: S^{+} and S^{−} have a real form. If they are quaternionic, after extension of scalars to ℂ, one becomes 2S^{+}, the other becomes 2S^{−}. In which case we are depends only on the signature p − q mod 8. Table:

p − q  real, complex or quaternionic
1 or 7  ℝ
2 or 6  ℂ
3 or 5  ℍ
4  ℍ, ℍ
8  ℝ, ℝ

3. Minkowski signature.

In Minkowski signature, (+, −, −, …), and with a positive light cone chosen on V, if S is an irreducible spinorial representation, there is up to a real factor a unique symmetric bilinear maps B: S ⊗ S → V. It can be normalized so that Q(s) := \frac{1}{2} B(s, s) is with values in the closed positive cone. Such a Q is now unique up to a positive real factor.

If S is complex, B is the real part of an hermitian bilinear form with values in V_{ℂ}: if J is the complex structure, Q is invariant by the J, as well as by the circle group of exp(θJ). After extension of scalars to ℂ, S becomes the sum of the two semi-spinorial representations, and B corresponds to a pairing S^{+} ⊗ S^{−} → V.

If S is quaternionic, the quaternions with absolute value one preserve Q. After extension of scalors to ℂ, S becomes the sum S_{0} ⊗T of two copies of a spinorial or semi-spinorial representation S_{0}, and B the tensor product of an antisymmetric pairing S_{0} ⊗ S_{0} → V_{ℂ} with a scalar alternating form T ⊗ T → ℂ. The quaternions become End(T).

The next table gives the nature of irreducible spinorial representations, as a function of n mod 8, for Spin(1, n−1). If Spin(1, n−1) has two irreducible spinorial representations, they have isomorphic restrictions to Spin(1, n − 2). The last column of the table describes the restriction of an irreducible representation of Spin(1, n−1) to Spin(1, n−2). It is either an irreducible spinorial representation, or twice it, or the sum of two distinct spinorial representations. Noted: S, 2S, S^{+} + S^{−}.

n mod 8  nature  restriction to SO(1, n − 2)
1  ℝ  S
2  ℝ, ℝ  S
(S^{+} and S^{−}, in duality)
3  ℝ  S^{+} + S^{−}
4  ℂ  2S
5  ℍ  2S
6  ℍ, ℍ  S
(S^{+} and S^{−}, in duality)
7  ℍ  S^{+} + S^{−}
8  ℂ  S