MATRIX GENERATORS FOR THE ORTHOGONAL GROUPS

L. J. RYLANDS AND D. E. TAYLOR

1. Introduction

Generators for the groups SL(l, q), Sp(2m, q), U(l, q) and Sz(q) have been available in computer algebra systems for some time [1, 4, 12]. Until recently it has only been practical to work with these groups for small dimensions and small fields. This covered small orthogonal groups (but not in their natural representation) because the orthogonal groups up to dimension 6 are isomorphic to other linear groups. For a complete description of the isomorphisms see [13, Chapters 11 and 12].

However, recent advances in computing speed and memory, as well as better algorithms, make it possible to work with larger groups. Hence there has been an increasing need for matrix generators for the orthogonal groups, particularly in dimensions beyond 6. This demand comes from several sources. For example, those working directly with orthogonal groups as well as those wishing to test new linear group recognition algorithms [10, 3] now need generators for all classical groups. The recent paper of Ishibashi and Earnest [7, 8] provides generators for O(l, q), but not for SO(l, q) nor its derived group Ω(l, q). The matrices of Ishibashi and Earnest have been implemented in GAP by Celler.

In 1962 Steinberg gave pairs of generators for all finite simple groups of Lie type. Steinberg’s generators are given in terms of root elements and generators for the Weyl group. In this paper we describe the corresponding generators for the finite orthogonal groups Ω(l, q). These generators are presented as matrices and are equal to Steinberg’s generators modulo the centre of the group. The purpose is to provide explicit constructions for the orthogonal groups which can be used within computer algebra packages such as Magma or GAP [1, 4]. Our methods are easily adapted to provide generators for SO(l, q) and O(l, q).

In the first part of the paper we outline the (well known) connection between the orthogonal groups and the Chevalley groups of types B_{m}, D_{m} and ^{2}D_{m} via their Lie algebras. This leads directly to the construction of the matrix generators in terms of root elements and the BN-pair structure.

Date: 5 March 1997.

2. Preliminaries

There are three families of finite orthogonal groups and, except for a few small cases, they correspond to Dynkin diagrams of types B_{m}, D_{m} and ^{2}D_{m}. Steinberg [11] provided generators for the Chevalley groups in terms of root elements and it is our intention to lift his generators to the corresponding matrix groups. As a general reference see [5], particularly sections 8 and 25.2.

We begin by reviewing the connection between orthogonal groups defined by quadratic forms and Chevalley groups defined in terms of simple Lie algebras of types B_{m} and D_{m}.

A quadratic form on a vector space V over a field 𝔽 is a function Q : V → 𝔽 such that Q(av) = a^{2}Q(v) and β(u, v) = Q(u+v)−Q(u)−Q(v) is bilinear. It is non-degenerate if Q(v) ≠ 0 for all non-zero elements v of the radical of β. For details see [13]. If the characteristic of 𝔽 is not 2, β is a non-degenerate symmetric bilinear form and uniquely determines Q. If the characteristic of 𝔽 is 2 then β is an alternating form and the dimension of its radical is either 1 or 0.

It is always possible to write V as an orthogonal direct sum

V = L_{1} ⊥ L_{2} ⊥ ⋯ ⊥ L_{m} ⊥ W

where L_{i} = ⟨e_{i}, f_{i}⟩, Q(e_{i}) = Q(f_{i}) = 0, β(e_{i}, f_{i}) = 1 and W is a subspace with no singular vectors. If 𝔽 is finite, the dimension of W is 0, 1 or 2. If 𝔽 is algebraically closed (e.g., ℂ) the dimension of W is 0 or 1. The integer m is the Witt index of Q and we let l be the dimension of V.

In describing the groups and Lie algebras associated with a quadratic form Q we shall write matrices with respect to the ordered basis

e_{1}, e_{2}, …, e_{m}, w_{1}, …, w_{k}, f_{m}, …, f_{1}

where w_{1}, …, w_{k} is a basis of W to be chosen later. We use J to denote the matrix of β with respect to this basis.

In the case 𝔽 = ℂ, the complex Lie algebra L of Q consists of all l ×l matrices X such that X^{t}J + JX = 0 with Lie product [xy] = xy − yx.

Over the complex numbers there is just one non-degenerate quadratic

form in each dimension. If l = 2m, we have J =(\begin{matrix}& & & & & 1 \\ & & . & . & . \\ & 1 \\ 1 \end{matrix}) and L is of type D_{m}. If l = 2m + 1, we may take W = ⟨w⟩, where Q(w) = 1

(hence β(w, w) = 2), J =(\begin{matrix}& & & & & & & & & & 1 \\ & & & & & & & . & . & . \\ & & & & & & 1 \\ & & & & & 2 \\ & & & & 1 \\ & . & . & . \\ 1 \end{matrix}) and L is of type B_{m}.

In type D_{m} the diagonal matrices of L have the form

h = diag(λ_{1}, λ_{2}, …, λ_{m}, −λ_{m}, …, −λ_{1})

and in type B_{m} they have the form

h = diag(λ_{1}, λ_{2}, …, λ_{m}, 0, −λ_{m}, …, −λ_{1}).

In both cases the set H of these matrices is a Cartan subalgebra.

The root space of α ∈ H^{∗} is L_{α} = {x ∈ L | [hx] = α(h)x} and the set Φ of non-zero α such that L_{α} ≠ 0 is the root system of H. For α ∈ Φ, dim L_{α} = 1 and we have the Cartan decomposition L = H ⊕ ⨁_{α∈Φ} L_{α}.

The real vector space spanned by the roots may be identified with the Euclidean space ℝ^{m} with orthonormal basis ε_{1}, ..., ε_{m} given by ε_{i}(h) = λ_{i}, where h is defined as above.

For each α ∈ Φ there is a unique element h_{α} ∈ H such that α(h_{α}) = 2 and h_{α} ∈ [L_{α}L_{−α}]. We shall choose a set of fundamental roots Δ = {α_{1}, α_{2}, …, α_{m}} and elements x_{α} ∈ L_{α}, for α ∈ Φ, such that {x_{α}, h_{α_{i}} | α ∈ Φ, 1 ≤ i ≤ m} forms a Chevalley basis for L. That is, [x_{α}x_{−α}] = h_{α} and for α, β, α + β ∈ Φ, [x_{α}x_{β}] = c_{α,β}x_{α+β}, where c_{α,β} = −c_{−α,−β} ∈ ℤ.

Let E_{ij} be the l × l matrix with 1 in the i, j-th position and 0 elsewhere; also let i′ denote l + 1 − i.

2.1. Case 1. Type D_{m} (l = 2m, m > 1). The set of roots is Φ = {±(ε_{i} ± ε_{j}) | 1 ≤ i < j ≤ m}. As fundamental roots for D_{m} we take

α_{1} = ε_{m−1} + ε_{m}, α_{2} = ε_{m−1} − ε_{m}, …, α_{m} = ε_{1} − ε_{2} corresponding to the Dynkin diagram

•\ •/•—•—•—⋯ —•

The set of positive roots is {ε_{i} ± ε_{j} | i < j}.

The elements h_{i} = h_{α_{i}} of the Chevalley basis are

h_{1} = E_{m−1,m−1} + E_{mm} − E_{m′m′} − E_{(m−1)′,(m−1)′}  and
h_{k} = E_{m+1−k,m+1−k} − E_{m+2−k,m+2−k}

+ E_{(m+2−k)′,(m+2−k)′} − E_{(m+1−k)′,(m+1−k)′} k = 2, …, m.

Then {h_{i} | 1 ≤ i ≤ m} is a basis for H.

Now define a map μ : Φ → L such that μ(α) spans L_{α} by

μ(ε_{i} − ε_{j}) = E_{ij} − E_{j′i′}  i ≠ j
μ(ε_{i} + ε_{j}) = −E_{ij′} + E_{ji′}  i < j
μ(−ε_{i} − ε_{j}) = E_{i′j} − E_{j′i} i < j.

The set μ(Φ) completes the Chevalley basis as is shown by the easy but tedious calculation that for all α, β ∈ Φ

[μ(α)μ(β)] = \begin{cases}±(r + 1)μ(α + β) & α + β ∈ Φ \\ h_{α} & α + β = 0 \\ 0 & α + β ∉ Φ ∪ \{0\} \end{cases}

where β − rα, …, β + qα is the α-string through β [5, §8.4]. Thus the Chevalley basis for L is

{μ(α) | α ∈ Φ} ∪ {h_{i} | 1 ≤ i ≤ m}.

If α is a positive root then μ(α) is an upper triangular matrix.

2.2. Case 2. Type B_{m} (l = 2m + 1). The set of roots is Φ = {±ε_{i}, ±(ε_{i} ± ε_{j}) | 1 ≤ i < j ≤ m}. As fundamental roots for B_{m} we take

α_{1} = ε_{m}, α_{2} = ε_{m−1} − ε_{m}, α_{3} = ε_{m−2} − ε_{m−1}, …, α_{m} = ε_{1} − ε_{2} corresponding to the Dynkin diagram

•— —•—•—⋯ —•

The positive roots are all ε_{i}, ε_{i} + ε_{j} and ε_{i} − ε_{j} (i < j).

The elements h_{i} = h_{α_{i}} of the Chevalley basis are

h_{1} = 2E_{mm} − 2E_{m′m′}  and
h_{k} = E_{m+1−k,m+1−k} − E_{m+2−k,m+2−k}

+ E_{(m+2−k)′,(m+2−k)′} − E_{(m+1−k)′,(m+1−k)′} k = 2, …, m.

Then {h_{i} | 1 ≤ i ≤ m} is a basis for H.

Now define a map μ : Φ → L such that μ(α) spans L_{α} by

μ(ε_{i} − ε_{j}) = E_{ij} − E_{j′i′}  i ≠ j
μ(ε_{i} + ε_{j}) = −E_{ij′} + E_{ji′}  i < j
μ(−ε_{i} − ε_{j}) = E_{i′j} − E_{j′i} i < j
μ(ε_{i}) = 2E_{i,m+1} − E_{m+1,i′}
μ(−ε_{i}) = −2E_{i′,m+1} + E_{m+1,i}.

The set μ(Φ) completes the Chevalley basis as is shown by the easy but tedious calculation that for all α, β ∈ Φ

[μ(α)μ(β)] = \begin{cases}±(r + 1)μ(α + β) & α + β ∈ Φ \\ h_{α} & α + β = 0 \\ 0 & α + β ∉ Φ ∪ \{0\} \end{cases} .

Thus the Chevalley basis for L is

{μ(α) | α ∈ Φ} ∪ {h_{i} | 1 ≤ i ≤ m}.

If α is a positive root then μ(α) is an upper triangular matrix.

The matrices of \frac{1}{k!}μ(α)^{k}, α ∈ Φ have integer entries [5, §25] and therefore exp(tμ(α)) may be interpreted over any field. For each prime power q we let L(q) denote the Lie algebra spanned by μ(Φ) over 𝔽_{q}. The algebra L(q) acts on the vector space V (q) and we shall use the same notation for its basis as for V. The quadratic form Q takes integral values on the integral linear combinations of this basis and so induces a quadratic form Q(q) on V (q). In all cases it remains nonsingular.

3. Orthogonal Groups

From now on we work with a fixed finite field 𝔽_{q} and abbreviate L(q), V (q) and Q(q) to L, V and Q. The group of all non-singular linear transformations of V which preserve Q is the orthogonal group O(V, Q); the intersection of the kernels of the spinor norm and the Dickson invariant is Ω(V, Q). When l = 2m and Q is a form of (maximal) Witt index m we denote this group by Ω^{+}(l, q) (L has type D_{m} in this case). When l = 2m+1 and the Witt index of Q is m we denote it by Ω^{0}(l, q) (L has type B_{m}). When l = 2m and the Witt index of Q is m − 1 we denote it by Ω^{−}(l, q) (see section 5). We also denote the groups Ω^{+}(l, q), Ω^{0}(l, q) and Ω^{−}(l, q), by Ω^{ϵ}(l, q) for ϵ = 1, 0, −1. Except for Ω^{+}(4, 2), Ω(V, Q) is the derived group of O(V, Q) [13, Theorems 11.45 and 11.51].

For l ≥ 3, the groups Ω^{ϵ}(l, q) are closely related to the Chevalley groups of adjoint type of the Lie algebra L of the previous section. The groups that do not arise as Chevalley groups are considered later.

The previous section describes a representation ϕ : L → 𝔤𝔩(l, q) of L as l × l matrices. For each μ(α) the matrix ϕ(μ(α)) is nilpotent, hence the sum

exp(ϕ(tμ(α))) = 1 + tϕμ(α) + \frac{t^{2}}{2!}ϕμ(α)^{2} + …

is finite (in fact in our case it has no more than 3 terms). Let x_{α}(t) = exp(ϕ(tμ(α))); then the Chevalley group associated with L and ϕ over 𝔽_{q} is

G = ⟨x_{α}(t) | t ∈ 𝔽_{q}, α ∈ Φ⟩.

The adjoint representation of L is the map ad : L → End(L) where ad(x)y = [xy]. It is not difficult to show [2, 4.5.1] that ad μ(α) is nilpotent. Let x̂_{α}(t) = exp(ad μ(α)), then [2, 4.5.1]

ϕ(x̂_{α}(t).z) = x_{α}(t)ϕ(z)x_{α}(t)^{−1} for z ∈ L.

Hence the group G is related to the Chevalley group of adjoint type, Ĝ = ⟨x̂_{α}(t) | t ∈ 𝔽_{q}, α ∈ Φ⟩, by the homomorphism θ : G → Ĝ where

θ(g).z = ϕ^{−1} (gϕ(z)g^{−1}) for z ∈ L, g ∈ G.

By construction θ(x_{α}(t)) = x̂_{α}(t) and therefore θ is onto. Every element in the kernel of θ commutes with the image of ϕμ, and hence the kernel of θ consists of scalar matrices.

Given a singular vector u and v ∈ ⟨u⟩^{⊥} we define a Siegel transformation to be a map ρ_{u,v} ∈ Ω(V) such that

ρ_{u,v}(x) = x + β(x, v)u − β(x, u)v − Q(v)β(x, u)u.

The ρ_{u,v} generate Ω(V) [13, Theorem 11.46]. For each root α, x_{α}(t) is a Siegel transformation. For example when α = ε_{i} − ε_{j}, x_{α}(t) = ρ_{te_{i},f_{j}} = I + t(E_{ij} − E_{j′i′}) and when α = ε_{i}, x_{α}(t) = ρ_{te_{i},w} = I + t(2E_{i,m+1} − E_{m+1,i′}) − t^{2}E_{ii′}.

This proves

Theorem 3.1. For ϵ = 0, 1, G = Ω^{ϵ}(l, q) and the kernel of θ has order 1 or 2.

4. Generators for Ω^{+}(l, q) and Ω^{0}(l, q)

In this section ξ will be a generator of the group 𝔽_{q}^{×}. For each positive root α there exists an element h_{α,ξ} such that h_{α,ξ}x_{β}(k)h_{α,ξ}^{−1} =

x_{β}(ξ^{2 \frac{(β,α)}{(α,α)}} k) for all k ∈ 𝔽_{q}^{×} and β ∈ Φ [11, Theorem 3.4]; in what follows a diagonal matrix will suffice, and it is easy to check that it satisfies the conditions.

As in [11, Theorem 3.7], for each α ∈ Φ^{+}, n_{α} = x_{α}(1)x_{−α}(−1)x_{α}(1). Write n_{i} for n_{α_{i}}.

We are now ready to give matrices for Steinberg’s generators for the groups Ω^{ϵ}(l, q) for ϵ = 0, 1.

4.1. Type D_{m} (l = 2m) m even m > 3. In this case

x_{α_{1}}(1) = I + ϕμ(α_{1}) = I − E_{m−1,m′} + E_{m,(m−1)′},
x_{−α_{1}}(1) = I + ϕμ(−α_{1}) = I + E_{(m−1)′,m} − E_{m′,m−1},
x_{α_{3}}(1) = I + ϕμ(α_{3}) = I + E_{m−2,m−1} − E_{(m−1)′,(m−2)′}.

The matrix (n_{jk}^{(1)}) of n_{1} = x_{α_{1}}(1)x_{−α_{1}}(−1)x_{α_{1}}(1) has 1’s on the diagonal and 0’s elsewhere except

n_{jk}^{(1)} = \begin{cases}0 & j = k = m − 1, m, m′ or (m − 1)′ \\ −1 & (j, k) = (m − 1, m′) or ((m − 1)′, m) \\ 1 & (j, k) = (m, (m − 1)′) or (m′, m − 1). \end{cases}

For i = 2, …, m, n_{i} = (n_{jk}^{(i)}) has 1’s on the diagonal and 0’s elsewhere except

n_{jk}^{(i)} = \begin{cases}0 & j = k = m + 1 − i, m + 2 − i, (m + 2 − i)′ or (m + 1 − i)′ \\ −1 & (j, k) = (m + 2 − i, m + 1 − i) \\ −1 & (j, k) = ((m + 2 − i)′, (m + 1 − i)′) \\ 1 & (j, k) = (m + 1 − i, m + 2 − i) \\ 1 & (j, k) = ((m + 1 − i)′, (m + 2 − i)′). \end{cases}

The matrix (n_{jk}) of n = n_{1}n_{2} … n_{m} has nonzero entries

n_{jk} = \begin{cases}(−1)^{m} & (j, k) = ((m − 1)′, 1) or (m − 1, 1′) \\ 1 & (j, k) = (m, m′) or (m′, m) \\ 1 & (j, k) = (i, i + 1) or (i′, (i + 1)′) for i = 1, …, m − 2. \end{cases}

The matrix (h_{jk}) of h_{α_{2},ξ} is the identity except that h_{m−1,m−1} = h_{m′,m′} = ξ and h_{m,m} = h_{(m−1)′,(m−1)′} = ξ^{−1}.

It follows from Steinberg [11, Theorem 3.13] that Ω^{+}(l, q) is generated by

\begin{cases}h_{α_{2},ξ} and x_{−α_{1}}(1)x_{α_{3}}(1)n & for q > 2 \\ x_{α_{1}}(1)x_{α_{3}}(1) and n & for q = 2 \end{cases} .

4.2. Type D_{m} (l = 2m) m odd m > 2. It follows from Steinberg [11, Theorem 3.11] that Ω^{+}(l, q) is generated by

\begin{cases}h_{α_{1},ξ} and x_{α_{1}}(1)n & for q > 3, \\ x_{α_{1}}(1) and n & for q = 2, 3 \end{cases}

where x_{α_{1}} and n are as in 4.1 and the matrix (h_{jk}) of h_{α_{1},ξ} is the identity except that h_{m−1,m−1} = h_{m,m} = ξ and h_{m′,m′} = h_{(m−1)′,(m−1)′} = ξ^{−1}.

4.3. Type D_{2} (l = 4). In this case the group is the central product SL(2, q) ∘ SL(2, q). Let p be the characteristic of 𝔽_{q}. We choose elements x (of order p) and z (of order q+1) which generate SL(2, q). (See [9, p.209] for a discussion of this point.) Since x and z have coprime order, SL(2, q) ∘ SL(2, q) is generated by (x, z) and (z, x).

To define z we take the underlying space for SL(2, q) to be the additive group of the field 𝔽_{q^{2}}. Then z is multiplication by ξ = ζ^{q−1}, where ζ is a primitive element of 𝔽_{q^{2}}. The element ξ has order q + 1 and satisfies a quadratic equation over 𝔽_{q} whose roots are ξ and ξ^{q} = ξ^{−1}. Thus the minimal polynomial for ξ is X^{2} − aX + 1, where a = ξ + ξ^{−1} and the matrix of z with respect to the basis 1, ξ for 𝔽_{q^{2}} is (\begin{matrix}0 & −1 \\ 1 & a \end{matrix}).

Taking x = (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}), it follows from Dickson’s Theorem [6, Hauptsatz

8.27] that x and z generate SL(2, q).

In order to represent the elements of SL(2, q) ∘ SL(2, q) as 4 × 4 matrices in Ω^{+}(4, q) we use the following construction. The group SL(2, q) × SL(2, q) acts on the space V of 2 × 2 matrices M over 𝔽_{q} such that (A, B).M = AMB^{t}. The determinant of M is a quadratic form of Witt index 2 preserved by this action and the image of SL(2, q) × SL(2, q) is isomorphic to SL(2, q) ∘ SL(2, q) and coincides with Ω^{+}(4, q). Using this repesentation Ω^{+}(4, q) is gener-

ated by the matrices (\begin{matrix}0 & −1 & 0 & −1 \\ 1 & a & −1 & a \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & a \end{matrix}) and (\begin{matrix}0 & 0 & 1 & −1 \\ 0 & 0 & 0 & −1 \\ −1 & −1 & a & −a \\ 0 & 1 & 0 & a \end{matrix}) corre-

sponding to (x, z) and (z, x) respectively, with respect to the basis

\{(\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) , (\begin{matrix}0 & −1 \\ 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 \\ 0 & 1 \end{matrix})\}

4.4. Type D_{1} (l = 2). In this case Ω^{+}(2, q) is the derived group of the full orthogonal group. It is cyclic, of order \frac{q−1}{2} when q is odd and

q − 1 when q is even. In both cases it is generated by (\begin{matrix}ξ^{2} & 0 \\ 0 & ξ^{−2} \end{matrix}).

4.5. Type B_{m} (l = 2m + 1) m ≥ 2. In this case h = h_{ε_{1}+ε_{m},ξ} is the identity except that h_{1,1} = h_{m,m} = ξ and h_{m′,m′} = h_{1′,1′} = ξ^{−1}. The matrix (n_{jk}^{(1)}) of n_{1} has 1’s on the diagonal and 0’s elsewhere except

n_{jk}^{(1)} = \begin{cases}0 & j = k = m or m′ \\ −1 & (j, k) = (m, m′), (m + 1, m + 1) or (m′, m). \end{cases}

For i = 2, …, m the matrix (n_{jk}^{(i)}) of n_{i} has 1’s on the diagonal and 0’s elsewhere except

n_{jk}^{(i)} = \begin{cases}0 & j = k = m + 1 − i, m + 2 − i, (m + 2 − i)′ or (m + 1 − i)′ \\ −1 & (j, k) = (m + 2 − i, m + 1 − i) \\ −1 & (j, k) = ((m + 2 − i)′, (m + 1 − i)′) \\ 1 & (j, k) = (m + 1 − i, m + 2 − i) \\ 1 & (j, k) = ((m + 1 − i)′, (m + 2 − i)′). \end{cases}

Then n = n_{1}n_{2} … n_{m} has nonzero entries

n_{jk} = \begin{cases}(−1)^{m} & (j, k) = (m′, 1) or (m, 1′) \\ −1 & (j, k) = (m + 1, m + 1) \\ 1 & (j, k) = (i, i + 1) or (i′, (i + 1)′) for i = 1, …, m − 1. \end{cases}

For odd q it follows from Steinberg [11, Theorem 3.11] that Ω^{0}(l, q) is generated by

\begin{cases}h and x_{α_{1}}(1)n & for q > 3 \\ x_{α_{1}}(1) and n & for q = 3 \end{cases}

where x_{α_{1}}(1) = I +ϕμ(α_{1})+ \frac{1}{2}ϕμ(α_{1})^{2} = I +2E_{m,m+1}−E_{m+1,m′} −E_{m,m′}.

For q even it follows from Steinberg [11, Theorem 3.14] that Ω^{0}(l, q) is generated by

\begin{cases}h and x_{ε_{1}−ε_{m}}(1)x_{−α_{1}}(1)n & for q > 2 \\ x_{ε_{1}−ε_{m}}(1)x_{−α_{1}}(1) and n & for q = 2 \end{cases}

where x_{ε_{1}−ε_{m}}(1) = I +E_{1,m} +E_{m′,1′} and x_{−α_{1}}(1) = I +E_{m+1,m} +E_{m′,m}.

4.6. Type B_{1}. The group Ω^{0}(3, q) is isomorphic to P SL(2, q) [13]. Consequently the Steinberg generators for P SL(2, q) provide generators for Ω^{0}(3, q). The group is generated by

\begin{cases}nx and h & for q > 3 \\ x and n & for q = 2, 3 \end{cases}

where n =(\begin{matrix}0 & 0 & −1 \\ 0 & −1 & 0 \\ −1 & 0 & 0 \end{matrix}), x =(\begin{matrix}1 & 0 & 0 \\ 1 & 1 & 0 \\ −1 & −2 & 1 \end{matrix}) and h =(\begin{matrix}ξ^{−2} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & ξ^{2} \end{matrix}).

5. Generators for Ω^{−}(l, q)

The group Ω^{−}(l, q) of type ^{2}D_{m} can be considered as a subgroup of the group G = Ω^{−}(l, q^{2}) of type D_{m+1}. Thus Ω^{−}(l, q) is a group of l × l matrices where l = 2m + 2.

The Dynkin diagram for type D_{m+1} has an automorphism of order 2: swap roots α_{1} and α_{2}, and leave the other fundamental roots fixed. This extends to an automorphism of the whole root system. Write \overline{α} for the image of α. For ^{2}D_{m} take the roots to be the orbits of this automorphism on the roots for type D_{m+1}. So the roots are

{1, 2}, {3}, …, {m}.

The Dynkin diagram for ^{2}D_{m} is •— —•—•—⋯ —•

The field 𝔽_{q^{2}} has an automorphism of order 2 given by t → t^{q}. Write \overline{t} for the image of t. The map given by x_{α}(t) → x\overline{α}(\overline{t}) extends to an automorphism of G of order 2. The groups Ω^{−}(l, q) are defined in terms of fixed points of this map. See Carter [2] for details.

Steinberg [11] gives a pair of generators for P Ω^{−}(l, q) in terms of elements of P Ω^{+}(l, q^{2}); we use this to get generators for Ω^{−}(l, q) in terms of elements of Ω^{+}(l, q^{2}). However, to get matrices with entries in the correct field (𝔽_{q}) we follow Carter [2, p. 271] and change the basis.

Let ν be a generator of 𝔽_{q^{2}} over 𝔽_{q}. In type D_{m+1} the quadratic form is Q(∑_{i=1}^{m+1}(a_{i}e_{i} + b_{i}f_{i})) = ∑_{i=1}^{m+1} a_{i}b_{i}. Take as a new basis e_{1}, …, e_{m}, w_{1}, w_{2}, f_{m}, …, f_{1} where e_{m+1} = w_{1}+νw_{2} and f_{m+1} = −w_{1}− \overline{ν}w_{2}. With respect to this new basis the quadratic form is given by Q′(∑_{i=1}^{m−1}(a_{i}e_{i} +b_{i}f_{i})+cw_{1} +dw_{2}) = ∑_{i=1}^{m−1}(a_{i}b_{i})+(d−νc)(d−\overline{ν}c)/(ν − \overline{ν})^{2}. Over the field 𝔽_{q} the quadratic form Q′ has Witt index m − 1.

With this new basis we can now write down Steinberg’s generators as l × l matrices with entries in 𝔽_{q}. The change of basis matrix is

S^{−1} =(\begin{matrix}I_{m} \\ & A \\ & & I_{m} \end{matrix}) where A =(\begin{matrix}1 & −1 \\ ν & −\overline{ν} \end{matrix}). Define h to be

S^{−1}h_{1,ν}h_{2,\overline{ν}}S = (\begin{matrix}I_{m−1} \\ & B \\ & & I_{m−1} \end{matrix})

with B =(\begin{matrix}ν\overline{ν} \\ & −1 & ν^{−1} + \overline{ν}^{−1} \\ & −ν − \overline{ν} & 1 + ν\overline{ν}^{−1} + ν^{−1}\overline{ν} \\ & & & ν^{−1}\overline{ν}^{−1} \end{matrix}), and x to be

S^{−1}x_{α_{1}}(1)\overline{x_{α_{1}}(1)}S = S^{−1}x_{α_{1}}(1)x_{α_{2}}(1)S = (\begin{matrix}I_{m−1} \\ & C \\ & & I_{m−1} \end{matrix})

with C =(\begin{matrix}1 & 1 & 0 & 1 \\ & 1 & 0 & 2 \\ & & 1 & ν + \overline{ν} \\ & & & 1 \end{matrix}).

Now n_{R} = x_{R}(1)x_{−R}(−1)x_{R}(1) (R is the new root {α_{1}, α_{2}}). With n_{3}, …, n_{m+1} as in type D_{m} (4.1) we calculate n = n_{R}n_{3}n_{4} … n_{m+1}.

The matrix (n_{jk}) of n has nonzero entries

n_{jk} = \begin{cases}(−1)^{m−1} & (j, k) = (m′, 1) or (m, 1′) \\ −1 & (j, k) = (m + 1, m + 1) \\ −ν − \overline{ν} & (j, k) = ((m + 1)′, m + 1) \\ 1 & (j, k) = ((m + 1)′, (m + 1)′) \\ 1 & (j, k) = (i, i + 1) or (i′, (i + 1)′) for i = 1, …, m − 1. \end{cases}

It follows from Steinberg that Ω^{−}(l, q) is generated by h and xn for all q.

6. Availability

The generators for Ω^{ϵ}(l, q) described in this paper as well as generators for SO^{ϵ}(l, q) and O^{ϵ}(l, q) have been included in Magma V2.10.

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L. J. Rylands Department of Mathematics The University of Western Sydney, Nepean

  E-mail address: l.rylands@nepean.uws.edu.au

D. E. Taylor School of Mathematics and Statistics University of Sydney N.S.W. 2006 Australia

  E-mail address: taylor d@maths.su.oz.au