Vector and Spin Representations#
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#
The Clifford group of \(C\) is
and the vector representation of \(\Gamma\) is the homomorphism \(\chi : \Gamma \to {\operatorname{GL}}(V)\) such that for \(s\in \Gamma\), \(\chi(s)\) is the linear transformation sending \(v\in V\) to \(s^{-1}vs\). In fact, \(\chi(\Gamma)\) 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 \({\bf F}_{2}\), if \(\dim V\) is even, \(\chi(\Gamma) = O(V,Q)\) and if \(\dim V\) is odd, \(\chi(\Gamma) = {\operatorname{SO}}(V,Q)\). (See [Chevalley, 1997] for further details.)
- VectorAction(g): AlgClffElt -> AlgMatElt#
The matrix of the Clifford group element \(g\in \Gamma\) acting on the quadratic space \(V\) by conjugation.
- Example: Clff Ref (ex-6e1ea1)#
If \(v\) is a non-singular element of \(V\), the negative of the linear transformation \(\chi(v)\) is the reflection in the hyperplane orthogonal to \(v\).
> 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
Siegel Transformations and Spin Groups#
The special Clifford group is \(\Gamma^+ = \Gamma\cap C_+\) and \(\chi(\Gamma^+) = {\operatorname{SO}}(V,Q)\). If \(\alpha\) is the main anti-automorphism of \(C\), the spin group of \(Q\) is
Given the polar form \(\beta\) of \(Q\) and linearly independent vectors \(u,v \in V\) such that \(u\) is singular and \(\beta(u,v) = 0\), the Siegel transformation defined by \(u\) and \(v\) (see Chapter Polar Spaces) is the isometry \(\rho_{u,v}\) defined by
In the Clifford algebra of \(Q\), the conditions on \(u\) and \(v\) become \(u^2 = 0\) and \(uv + vu = 0\). Thus \(uv - 1\in {\operatorname{Spin}}(V,Q)\) and \(\chi(uv-1) = \rho_{u,v}\). Except for \(\Omega^+(4,2)\) the group \(\Omega(V,Q)\) is generated by Siegel transformations and thus \(\chi :{\operatorname{Spin}}(V,Q) \to \Omega(V,Q)\) is onto; its kernel is \(\{\pm 1\}\).
- Example: siegel (ex-c93d8c)#
We verify that a Siegel transformation \(\rho_{u,v}\) can be obtained as the vector representation of \(uv-1\).
> 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
Regarding \(\Omega(V,Q)\) as a group of Lie type, the subgroups \(X_{u,v} = \langle\, \rho_{tu,v} \mid t \in F \,\rangle\) 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 \(\Omega(V,Q)\) are defined in terms of root elements (see [Rylands and Taylor, 1998]).
- Example: vecrep (ex-1c9644)#
When \(F = {\bf F}_{q}\) and \(\dim V = 2r\) we construct the elements of \({\operatorname{Spin}}(V,Q)\) which map onto the standard generators of \(\Omega^+(2r,q)\).
> q := 3; > r := 4; > K := GF(3); > Q := StandardQuadraticForm(2*r,K); > C,V,f := CliffordAlgebra(Q);
The root element \(x_{\alpha_k}(t)\) indexed by the \(k\)th simple root is given by 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 \({\operatorname{Spin}}(V,Q)\) is not faithful.
> VectorAction(-One(C)) eq IdentityMatrix(K,2*r); true
Spin Representations#
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 \(\Gamma\), \(\Gamma^+\) and \({\operatorname{Spin}}(V,Q)\) are also called spin representations. The spin representation of \(\Gamma\) is irreducible except when the field has order 2, \(r = 1\) and the Witt index is 1.
- ActionMatrix(S, s): AlgAss, AlgAssElt -> AlgMatElt#
The matrix representing the action of \(s\) on the right ideal \(S\) of an associative algebra \(A\) and \(s\) is an element of \(A\).
- Example: ideals (ex-b3def3)#
> 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]
- Example: spinrep (ex-d57a8d)#
The Magma functions
Spin,SpinPlusandSpinMinusconstruct the groups using highest weight representations of the corresponding group of Lie type. But in this example we construct the spin representation of \({\operatorname{Spin}}^+(8,3)\) by working directly with the Clifford algebra.> q := 3; > r := 4; > K := GF(3); > Q := StandardQuadraticForm(2*r,K); > C,V,f := CliffordAlgebra(Q);
We adapt the code from Example Example: vecrep.
The root element \(x_{\alpha_k}(t)\) indexed by the \(k\)th simple root is given by the Magma function
x(k,t)and the corresponding negative root element isy(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 \({\operatorname{Spin}}^+(8,3)\) 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