Throughout this section, let Q be a quadratic form of Witt index r on a vector space V over a finite field F such that the polar form of Q is non-degenerate and let C denote the Clifford algebra of Q.
The Clifford group of C is
Γ = { s ∈C | s is a unit and s - 1vs ∈V for all v∈V }
and the vector representation of Γ is the homomorphism χ : Γ to GL(V) such that for s∈Γ, χ(s) is the linear transformation sending v∈V to s - 1vs. In fact, χ(Γ) is a subgroup of the orthogonal group O(V, Q). Except when the Witt index of Q is 2 and V is a space of dimension 4 over GF(2), if dim V is even, χ(Γ) = O(V, Q) and if dim V is odd, χ(Γ) = SO(V, Q). (See [Che97] for further details.)
The matrix of the Clifford group element g∈Γ acting on the quadratic space V by conjugation.
> F := GF(3); > Q := StandardQuadraticForm(6,F); > C, V, f := CliffordAlgebra(Q); > v := V![2,0,2,0,0,1];
In order to ensure v is non-singular we check its quadratic norm.
> QuadraticNorm(v); 2 > A := VectorAction(f(v)); > -A eq OrthogonalReflection(v); true
The special Clifford group is Γ^ + = Γ∩C_ + and χ(Γ^ + ) = SO(V, Q). If α is the main anti-automorphism of C, the spin group of Q is
Spin(V, Q) = { s∈Γ^ + | α(s)s = 1 }.
Given the polar form β of Q and linearly independent vectors u, v ∈V such that u is singular and β(u, v) = 0, the Siegel transformation defined by u and v (see Chapter POLAR SPACES) is the isometry ρu, v defined by
x ρu, v = x + β(x, v)u - β(x, u)v - Q(v)β(x, u) u.
In the Clifford algebra of Q, the conditions on u and v become u2 = 0 and uv + vu = 0. Thus uv - 1∈Spin(V, Q) and χ(uv - 1) = ρu, v. Except for Ω^ + (4, 2) the group Ω(V, Q) is generated by Siegel transformations and thus χ :Spin(V, Q) to Ω(V, Q) is onto; its kernel is {∓ 1}.
Regarding Ω(V, Q) as a group of Lie type, the subgroups Xu, v = < ρtu, v | t ∈F > are root groups. (If dim V = 2r, we consider only the groups for which both u and v are singular.) In Magma, the generators of the groups Ω(V, Q) are defined in terms of root elements (see [RT98]).
> F := GF(3); > Q := StandardQuadraticForm(6,F); > C, V, f := CliffordAlgebra(Q); > u := V.1; > v := V.5; > VectorAction(f(u)*f(v) - One(C)) eq SiegelTransformation(u,v); true
> q := 3; > r := 4; > K := GF(3); > Q := StandardQuadraticForm(2*r,K); > C,V,f := CliffordAlgebra(Q);
The root element xαk(t) indexed by the kth simple root is given be the following function.
> x := func< k,t | > k eq 1 select VectorAction(f(t*V.(r+2))*f(V.(r+1))-One(C)) > else VectorAction(f(t*V.(r-k+2))*f(V.(r+k))-One(C)) >;
It turns out that our choice of Q ensures that the matrices of the negative root elements are the transposed matrices of the corresponding positive roots.
> n := func< k, t | x(k,t)*Transpose(x(k,-t^-1))*x(k,t) >; > h := func< k, t | n(k,t)*n(k,-1) >; > w := n(1,1)*n(2,1)*n(3,1)*n(4,1); > xi := PrimitiveElement(K); > G := OmegaPlus(2*r,q); > G.1 eq h(2,xi); true > G.2 eq Transpose(x(1,1))*x(3,1)*w; true
Note that the vector representation of Spin(V, Q) is not faithful.
> VectorAction(-One(C)) eq IdentityMatrix(K,2*r); true
If the dimension of V is 2r, the Clifford algebra C of Q is simple and hence all irreducible representations are equivalent. We may take the representation space to be a minimal right ideal S of C. The elements of S are spinors and the representation itself is the spin representation. The restrictions of this representation to the groups Γ, Γ^ + and Spin(V, Q) are also called spin representations. The spin representation of Γ is irreducible except when the field has order 2, r = 1 and the Witt index is 1.
The matrix representing the action of s on the right ideal S of an associative algebra A and s is an element of A.
> F := GF(5); > Q := StandardQuadraticForm(4,F); > C,V,f := CliffordAlgebra(Q); > E, h := EvenSubalgebra(C); > IsSimple(E); false > S := MinimalRightIdeals(E)[1]; > s := (f(V.1+V.4)*f(V.2+V.3))@@h; > ActionMatrix(S,s); [0 4] [1 0]
> q := 3; > r := 4; > K := GF(3); > Q := StandardQuadraticForm(2*r,K); > C,V,f := CliffordAlgebra(Q);We adapt the code from Example H95E12.
The root element xαk(t) indexed by the kth simple root is given be the Magma function x(k,t) and the corresponding negative root element is y(k,t).
> x := func< k,t | > k eq 1 select f(t*V.(r+2))*f(V.(r+1))-One(C) > else f(t*V.(r-k+2))*f(V.(r+k))-One(C) >; > y := func< k,t | > k eq 1 select f(t*V.r)*f(V.(r-1))-One(C) > else f(t*V.(r-k+1))*f(V.(r+k-1))-One(C) >;
The other functions are the same as before.
> n := func< k, t | x(k,t)*y(k,-t^-1)*x(k,t) >; > h := func< k, t | n(k,t)*n(k,-1) >; > w := n(1,1)*n(2,1)*n(3,1)*n(4,1);
The spin representation space is a minimal right ideal of the Clifford algebra.
> S := MinimalRightIdeals(C : Limit := 1)[1]; > Dimension(S); 16 > X := sub<GL(16,K) | ActionMatrix(S,h(2,2)), ActionMatrix(S,y(1,1)*x(3,1)*w) >; > LieType(X,3); true <"D", 4, 3> > LMGOrder(X); 19808719257600 > Z := LMGCentre(X); > #Z, IsElementaryAbelian(Z); 4 true > SS := SpinPlus(8,K); > #SS; 19808719257600
The spin representation of Spin^ + (8, 3) is the direct sum of two half spin representations, neither of which is faithful. The half spin spaces are minimal ideals of the even subalgebra of C.
> E, phi := EvenSubalgebra(C); > T := MinimalRightIdeals(E : Limit := 1)[1]; > Dimension(T); 8 > Y := sub<GL(8,K) | ActionMatrix(T,h(2,2)@@phi), > ActionMatrix(T,(y(1,1)*x(3,1)*w)@@phi) >; > LieType(Y,3); true <"D", 4, 3> > LMGOrder(Y); 9904359628800