In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are representations of the spin groups, which are double covers of the special orthogonal groups. They are usually studied over the real or complex numbers, but they can be defined over other fields.

Elements of a spin representation are called spinors. They play an important role in the physical description of fermions such as the electron.

The spin representations may be constructed in several ways, but typically the construction involves (perhaps only implicitly) the choice of a maximal isotropic subspace in the vector representation of the group. Over the real numbers, this usually requires using a complexification of the vector representation. For this reason, it is convenient to define the spin representations over the complex numbers first, and derive real representations by introducing real structures.

The properties of the spin representations depend in a subtle way on the dimension and the signature of the orthogonal group. In particular, spin representations often admit invariant bilinear forms which can be used to embed the spin groups into classical Lie groups. In low dimensions, these embeddings are surjective and determine special isomorphisms between the spin groups and more familiar Lie groups; this elucidates the properties of spinors in these dimensions.

Set up

Let V be a finite dimensional real or complex vector space with a nondegenerate quadratic form Q. The (real or complex) linear maps preserving Q form the orthogonal group O(V,Q). The identity component of the group will be called the special orthogonal group SO(V,Q). (For V real with an indefinite quadratic form, this terminology is not standard: the special orthogonal group is usually defined to be a subgroup with two components in this case.) Up to group isomorphism, SO(V,Q) has a unique connected double cover, the spin group Spin(V,Q). There is thus a group homomorphism Spin(V,Q) ? SO(V,Q) whose kernel has two elements denoted {1, -1}, where 1 is the identity element.

O(V,Q), SO(V,Q) and Spin(V,Q) are all Lie groups, and for fixed (V,Q) they have the same Lie algebra, so(V,Q). If V is real, then V is a real vector subspace of its complexification V_C := V ?_R C, and the quadratic form Q extends naturally to a quadratic form Q_C on V_C. This embeds SO(V,Q) as a subgroup of SO(V_C, Q_C), and hence we may realise Spin(V,Q) as a subgroup of Spin(V_C, Q_C). Furthermore, so(V_C, Q_C) is the just the complexification of so(V,Q).

In the complex case, quadratic forms are determined up to isomorphism by the dimension n of V. Concretely, we may assume V=Cⁿ and

The corresponding Lie groups and Lie algebra are denoted O(n,C), SO(n,C), Spin(n,C) and so(n,C).

In the real case, quadratic forms are determined up to isomorphism by a pair of nonnegative integers (p,q) where n:=p+q is the dimension of V, and p-q is the signature. Concretely, we may assume V=Rⁿ and

The corresponding Lie groups and Lie algebra are denoted O(p,q), SO(p,q), Spin(p,q) and so(p,q). We write R^p,q in place of Rⁿ to make the signature explicit.

The spin representations are in some sense the simplest representations of Spin(n,C) and Spin(p,q) which do not come from representations of SO(n,C) and SO(p,q). A spin representation is therefore, in particular, a real or complex vector space S together with a group homomorphism ? from Spin(n,C) or Spin(p,q) to the general linear group GL(S) such that the element -1 is not in the kernel of ?.

If S is such a representation, then according to the relation between Lie groups and Lie algebras, it induces a Lie algebra representation, i.e., a Lie algebra homomorphism from so(n,C) or so(p,q) to the Lie algebra gl(S) of endomorphisms of S with the commutator bracket.

Spin representations can be analysed according to the following strategy: if S is a real spin representation of Spin(p,q), then its complexification is a complex spin representation of Spin(p,q); as a representation of so(p,q), it therefore extends to a complex representation of so(n,C). Proceeding in reverse, we therefore first construct complex spin representations of Spin(n,C) and so(n,C), then restrict them to complex spin representations of so(p,q) and Spin(p,q), then finally analyse possible reductions to real spin representations.

Complex spin representations

Let V=Cⁿ with the standard quadratic form Q so that

The symmetric bilinear form on V associated to Q by polarization will be denoted <.,.>.

Isotropic subspaces and root systems

A standard construction of the spin representations of so(n,C) begins with a choice of a pair (W, W^*) of maximal isotropic subspaces of V with W n W^* = 0. Let us make such a choice. If n = 2m or n = 2m+1, then W and W^* both have dimension m. If n = 2m, then V = W ? W^*, whereas if n = 2m+1, then V = W ? U ? W^*, where U is the 1-dimensional orthogonal complement to W ? W^*. The bilinear form <.,.> induces a pairing between W and W^* which must be nondegenerate, because W and W^* are isotropic subspaces and Q is nondegenerate. Hence W and W^* are dual vector spaces.

More concretely, let a₁, ... a_m be a basis for W. Then there is a unique basis a₁, ... a_m of W^* such that

If A is a m × m matrix, then A induces a endomorphism of W with respect to this basis and the transpose A^T induces a transformation of W^* with

for all w in W and w* in W^*. It follows that the endomorphism ?_A of V, equal to A on W, - A^T on W^* and zero on U (if n is odd), is skew

and hence an element of so(n,C).

Using the diagonal matrices in this construction defines a Cartan subalgebra h of so(n,C): the rank of so(n,C) is m, and the diagonal m × m matrices determine an m-dimensional abelian subalgebra.

Let e₁, ... e_m be the basis of h^* such that, for a diagonal matrix A, e_k(?_A) is the kth diagonal entry of A. Clearly this is a basis for h^*. Since the bilinear form identifies so(n,C) with , it is now easy to construct the root system associated to h. The root spaces (simultaneous eigenspaces for the action of h) are spanned by the following elements:

with root (simultaneous eigenvalue)

(which is in h if i = j) with root

with root

and, if n is odd, and u is a nonzero element of U,

with root

with root

Thus, with respect the basis e₁, ... e_m, the roots are the vectors in h^* which are permutations of

together with the permutations of

if n = 2m+1 is odd.

A system of positive roots is given by e_i+e_j (i?j), e_i-e_j (i<j) and (for n odd) e_i. The corresponding simple roots are

The positive roots are nonnegative integer linear combinations of the simple roots.

Spin representations and their weights

One construction of the spin representations of so(n,C) uses the exterior algebra(s)

and/or

There is an action of V on S such that for any element v = w+w* in W ? W^* and any ? in S the action is given by:

where the second term is a contraction (interior multiplication) defined using the bilinear form which pairs W and W^*. This action respects the Clifford relations v² = Q(v)1, and so induces a homomorphism from the Clifford algebra Cl_nC of V to End(S). A similar action can be defined on S', so that both S and S' are Clifford modules.

The Lie algebra so(n,C) is a Lie subalgebra of Cl_nC (under the commutator bracket) via the embedding

It follows that both S and S' are representations of so(n,C). They are actually equivalent representations, so we focus on S.

The explicit description shows that the elements a_i?a_i of the Cartan subalgebra h act on S by

A basis for S is given by elements of the form

for 0 = k = m and i₁ < ... < i_k. These clearly span weight spaces for the action of h: a_i?a_i has eigenvalue -1/2 on the given basis vector if i = i_j for some j, and has eigenvalue 1/2 otherwise.

It follows that the weights of S are all possible combinations of

and each weight space is one dimensional. Elements of S are called Dirac spinors.

When n is even, S is not an irreducible representation: and are invariant subspaces. The weights are divided into those which have an even number of minus signs, and those which have an odd number of minus signs. Both S₊ and S_- are irreducible representations of dimension 2^m-1 whose elements are called Weyl spinors. They are also known as chiral spin representations or half-spin representations. With respect to the positive root system above, the highest weights of S₊ and S_- are

and

respectively. The Clifford action identifies Cl_nC with End(S) and the even subalgebra is identified with the endomorphisms preserving S₊ and S_-. The other Clifford module S' is isomorphic to S in this case.

When n is odd, S is an irreducible representation of so(n,C) of dimension 2^m: the Clifford action of a unit vector u ? U is given by

and so elements of so(n,C) of the form u?w or u?w* do not preserve the even and odd parts of the exterior algebra of W. The highest weight of S is

The Clifford action is not faithful on S: Cl_nC can be identified with End(S) ? End(S'), where u acts with the opposite sign on S'. More precisely, the two representations are related by the parity involution a of Cl_nC (also known as the principal automorphism), which is the identity on the even subalgebra, and minus the identity on the odd part of Cl_nC. In other words, there is a linear isomorphism from S to S', which identifies the action of A in Cl_nC on S with the action of a(A) on S'.

Bilinear forms

if ? is a weight of S, so is -?. It follows that S is isomorphic to the dual representation S^*.

When n = 2m+1 is odd, the isomorphism B: S ? S^* is unique up to scale by Schur's lemma, since S is irreducible, and it defines a nondegenerate invariant bilinear form ß on S via

Here invariance means that

for all ? in so(n,C) and f, ? in S — in other words the action of ? is skew with respect to ß. In fact, more is true: S^* is a representation of the opposite Clifford algebra, and therefore, since Cl_nC only has two nontrivial simple modules S and S', related by the parity involution a, there is an antiautomorphism t of Cl_nC such that

for any A in Cl_nC. In fact t is reversion (the antiautomorphism induced by the identity on V) for m even, and conjugation (the antiautomorphism induced by minus the identity on V) for m odd. These two antiautomorphisms are related by parity involution a, which is the automorphism induced by minus the identity on V. Both satisfy t(?) = -? for ? in so(n,C).

When n = 2m, the situation depends more sensitively upon the parity of m. For m even, a weight ? has an even number of minus signs if and only if -? does; it follows that there are separate isomorphisms B_±: S_± ? S_±^* of each half-spin representation with its dual, each determined uniquely up to scale. These may be combined into an isomorphism B: S ? S^*. For m odd, ? is a weight of S₊ if and only if -? is a weight of S_-; thus there is an isomorphism from S₊ to S_-^*, again unique up to scale, and its transpose provides an isomorphism from S_- to S₊^*. These may again be combined into an isomorphism B: S ? S^*.

For both m even and m odd, the freedom in the choice of B may be restricted to an overall scale by insisting that the bilinear form ß corresponding to B satisfies (1), where t is a fixed antiautomorphism (either reversion or conjugation).

Symmetry and the tensor square

The symmetry properties of ß: S ? S ? C can be determined using Clifford algebras or representation theory. In fact much more can be said: the tensor square S ? S must decompose into a direct sum of k-forms on V for various k, because its weights are all elements in h^* whose components belong to {-1,0,1}. Now equivariant linear maps S ? S ? ?^kV^* correspond bijectively to invariant maps ?^kV ? S ? S ? C and nonzero such maps can be constructed via the inclusion of ?^kV into the Clifford algebra. Furthermore if ß(f,?) = e ß(?,f) and t has sign e_k on ?^kV then

for A in ?^kV.

If n = 2m+1 is odd then it follows from Schur's Lemma that

(both sides have dimension 2^2m and the representations on the right are inequivalent). Because the symmetries are governed by an involution t which is either conjugation or reversion, the symmetry of the ?^2jV^* component alternates with j. Elementary combinatorics gives

and the sign determines which representations occur in S²S and which occur in ?²S.^[1] In particular

and

for v ? V (which is isomorphic to ?^2mV), confirming that t is reversion for m even, and conjugation for m odd.

If n=2m is even, then the analysis is more involved, but the result is a more refined decomposition: S²S_±, ?²S_± and S₊ ? S_- can each be decomposed as a direct sum of k-forms (where for k=m there is a further decomposition into selfdual and antiselfdual m-forms).

The main outcome is a realisation of so(n,C) as a subalgebra of a classical Lie algebra on S, depending upon n modulo 8, according to the following table:

n mod 8	0	1	2	3	4	5	6	7
Spinor algebra

For n = 6, these embeddings are isomorphisms (onto sl rather than gl for n=6):

Real representations

The complex spin representations of so(n,C) yield complex representations S of so(p,q) by restricting the action to the real subalgebras. However, there are additional "reality" structures which are invariant under the action of the real Lie algebras. These come in three types.

1. There is an invariant complex antilinear map r: S ? S with r² = id_S. The fixed point set of r is then a real vector subspace S_R of S with S_R ? C = S. This is called a real structure.
2. There is an invariant complex antilinear map j: S ? S with j² = -id_S. It follows that the triple i, j and k:=ij make S into a quaternionic vector space S_H. This is called a quaternionic structure.
3. There is an invariant complex antilinear map b: S ? S^* which is invertible. This defines a hermitian bilinear form on S and is called a hermitian structure.

The type of structure invariant under so(p,q) depends only on the signature p-q modulo 8, and is given by the following table.

Here R, C and H denote real, hermitian and quaternionic structures respectively, and R+R and H+H indicate that the half-spin representations both admit real or quaternionic structures respectively.

Description and tables

To complete the description of real representation, we must describe how these structures interact with the invariant bilinear forms. Since n = p+q ? p - q mod 2, there are two cases: the dimension and signature are both even, and the dimension and signature are both odd.

The odd case is simpler, there is only one complex spin representation S, and hermitian structures do not occur. Apart from the trivial case n = 1, S is always even dimensional, say dim S = 2N. The real forms of so(2N,C) are so(K,L) with K + L = 2N and so*(N,H), while the real forms of sp(2N,C) are sp(2N,R) and sp(K,L) with K + L = N. The presence of a Clifford action of V on S forces K = L in both cases unless pq = 0, in which case KL=0, which is denoted simply so(2N) or sp(N). Hence the odd spin representations may be summarized in the following table.

(†) N is even for n>3 and for n=3, this is sp(1).

The even dimensional case is similar. For n>2, the complex half-spin representations are even dimensional. We have additionally to deal with hermitian structures and the real forms of sl(2N,C), which are sl(2N,R), su(K,L) with K + L = 2N, and sl(N,H). The resulting even spin representations are summarized as follows.

(*) For pq=0, we have instead so(2N)+so(2N)

(†) N is even for n>4 and for pq=0 (which includes n=4 with N=1), we have instead sp(N)+sp(N)

The low dimensional isomorphisms in the complex case have the following real forms.

Euclidean signature	Minkowskian signature	Other signatures

The only special isomorphism of real Lie algebras missing from this table is

References

• Brauer, Richard; Weyl, Hermann (1935), "Spinors in n dimensions", American Journal of Mathematics 57 (2): 425–449 .
• Cartan, Élie (1966), The theory of spinors, Paris, Hermann (reprinted 1981, Dover Publications), ISBN 978-0486640709 .
• Chevalley, Claude (1954), The algebraic theory of spinors and Clifford algebras, Columbia University Press (reprinted 1996, Springer), ISBN 978-3540570639 .
• Deligne, Pierre (1999), "Notes on spinors", in P. Deligne, P. Etingof, D. S. Freed, L. C. Jeffrey, D. Kazhdan, J. W. Morgan, D. R. Morrison, E. Witten, Quantum Fields and Strings: A Course for Mathematicians, Providence: American Mathematical Society, pp. 99–135 . See also the programme website for a preliminary version.
• Fulton, William; Harris, Joe (1991), Representation theory. A first course, Graduate Texts in Mathematics, Readings in Mathematics, 129, New York: Springer-Verlag, MR1153249, ISBN 0-387-97527-6, ISBN 0-387-97495-4 .
• Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0123296504 .
• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989), Spin Geometry, Princeton University Press, ISBN 0-691-08542-0 .
• Weyl, Hermann (1946), The Classical Groups: Their Invariants and Representations (2nd ed.), Princeton University Press (reprinted 1997), ISBN 978-0691057569 .

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