MATRIX GENERATORS FOR EXCEPTIONAL GROUPS OF LIE TYPE

R. B. HOWLETT, L. J. RYLANDS, AND D. E. TAYLOR

Abstract. This paper gives a uniform method of constructing generators for finite groups of Lie type with particular emphasis on the exceptional groups. The algorithm constructs matrices for the action of root elements on the lowest dimension representation of an associated Lie algebra. These generators have been implemented in the computer algebra system Magma and this completes the provision of pairs of matrix generators for all finite groups of Lie type.

1. Introduction

Until recently it was impractical to use general purpose computer algebra systems to investigate Chevalley groups except for those of small rank over small fields. But computer algebra systems (such as Magma (Bosma et al. 1997) and GAP (Schönert et al. 1994)) now have the power to deal with some aspects of all finite groups of Lie type. A natural way to represent these groups is via matrices over the defining field. Thus, for computational purposes, there is a need to provide matrix generators for these groups. This has been done for the classical groups (Taylor 1987, Rylands and Taylor 1998) and now it remains to extend this to all finite groups of Lie type.

It is the purpose of the present paper to give a uniform method of constructing generators for groups of Lie type with particular emphasis on the exceptional groups. The constructions described here could be carried out within any computer algebra system and, in particular, have been implemented in Magma. This completes the determination of matrix generators for all groups of Lie type, including the twisted groups of Steinberg, Suzuki and Ree (and the Tits group).

The Lie algebras and related Chevalley groups of types A_{n}, B_{n}, C_{n} and D_{n} can be identified with classical groups (Carter (1972), §11.3), and in Taylor (1987) and Rylands and Taylor (1998) this identification was used to translate the generators given by Steinberg (1962) to matrix forms.

The constructions in this paper rely on an investigation of the root systems of Lie algebras, providing a uniform approach and avoiding case by case discussions of nonassociative algebras. In each case (except E_{8}) we obtain the lowest dimensional module for the Lie algebra via an embedding in a Lie algebra of higher rank.

2. Modules

In this section we prove two theorems about Lie algebras and root systems for use in the remainder of the paper. For notation and general facts about Lie algebras see Carter (1972) or Humphreys (1972).

Date: 24 June 1999.

Let L denote a complex semisimple Lie algebra with root system Φ, fundamental roots Δ, Cartan subalgebra H and Cartan decomposition L = H ⊕⨁_{α∈Φ} L_{α}. For each α ∈ Φ choose e_{α} ∈ L_{α} so that { e_{α} | α ∈ Φ } together with the elements h_{α} = [e_{α} e_{−α}] for α ∈ Δ forms a Chevalley basis for L (Carter (1972), Theorem 4.2.1). In particular, { h_{α} | α ∈ Δ } is a basis for H, Φ is a subset of the dual space of H and [h e_{β}] = β(h) for all h ∈ H. We let Φ^{+} denote the set of positive roots with respect to Δ and put Φ^{−} = { −α | α ∈ Φ^{+} }. The height of a root is the sum of its coefficients when expressed as a sum of fundamental roots.

Let κ denote the Killing form on the dual of H. Then ⟨ α, β ⟩ = 2κ(α, β)/κ(β, β) is linear in the first variable and if Δ = {α_{1}, …, α_{ℓ}}, the i j-th entry of the Cartan matrix is ⟨ α_{i}, α_{j} ⟩.

When α + β ∈ Φ, write c_{α,β} for the structure constant given by [e_{α} e_{β}] = c_{α,β} e_{α+β}. For convenience, define c_{α,β} to be 0 when α + β is neither 0 nor a root.

As usual, the adjoint action of L on itself is defined by (ad x)y = [x y].

Theorem 1. Let L be a semisimple Lie algebra with root system Φ and fundamental roots Δ. Given a subset Δ_{1} of Δ, let Φ_{1} be the root system generated by Δ_{1} and let L_{1} be the corresponding Lie subalgebra of L. For γ ∈ Φ \ Φ_{1}, let

X = { α ∈ Φ | α − γ is in the span of Δ_{1} }.

Then the subspace V of L spanned by the e_{α} where α ∈ X is an L_{1}-module of dimension |X|.

Proof. The restriction of the adjoint action of L to L_{1} allows us to view L as an L_{1}- module. For α ∈ Φ_{1} and β ∈ X it is clear that α + β − γ is a linear combination of elements of Δ_{1}. Therefore, if α + β ∈ Φ, then α + β ∈ X and hence ad(e_{α})e_{β} ∈ V. To complete the proof that V is an L_{1}-submodule of L we note that we cannot have α + β = 0, otherwise −γ would belong to the span of Δ_{1}, contrary to the choice of γ.

Beginning with a Dynkin diagram for L_{1} and a finite field 𝔽, we give an algorithmic construction of an L_{1}-module V and of the matrices representing the action of exp(t ad(e_{α})) on V (with respect to the basis indexed by X of Theorem 1), where α ∈ Φ_{1} and t ∈ 𝔽.

In each case we label the fundamental roots with lower case letters a, b, c, …. This provides a natural total order ≺ on Δ and we suppose that Δ \ Δ_{1} = {ω} where ω is the last element of Δ. Let V be the L_{1}-module corresponding to γ = ω. We see from the proof of Theorem 1 that X consists of the roots in which the coefficient of ω is 1 when expressed as a linear combination of fundamental roots; in particular, X ⊆ Φ^{+}. The Dynkin diagram is conveniently described by its Cartan matrix and, given this information, the algorithm of Jacobson (1962), p. 122 (and sketched in §11.1 of Humphreys (1972)) constructs the positive roots of Φ such that roots of equal height are ordered lexicographically. We shall write α ≺ β to indicate that α comes before β in this order.

To obtain upper triangular matrices for the linear transformations ad(e_{α}) when α is positive, we order X, and hence the basis { e_{α} | α ∈ X } of V, according to the reverse of Jacobson’s ordering. Given α ∈ Φ_{1}, the matrix of the restriction of ad(e_{α}) to V has c_{α,β} in the row indexed by α + β and the column indexed by β, whenever α + β is a root.

The assumption that we are working with a Chevalley basis for L means that the c_{α,β} are determined up to a sign. In fact c_{α,β} = ±(r + 1) where r is the greatest integer such that β − rα is a root. In order to determine the sign we follow the method of Carter (1972), §4.2. For each positive root γ which is not a fundamental root we choose the least fundamental root α (in the total ordering ≺) such that β = γ − α is a root. In the terminology of Carter, (α, β) is an extraspecial pair. The signs of the structure constants c_{α,β} for extraspecial pairs (α, β) may be chosen arbitrarily and then all other structure constants are uniquely determined.

Given the ordered set of positive roots it is a straightforward matter to determine the extraspecial pairs. Then for roots α and β (α + β ≠ 0) we may use the following algorithm (derived from Carter (1972), Theorem 4.1.2) to determine the c_{α,β} recursively, given that we take c_{α,β} to be positive whenever (α, β) is an extraspecial pair. The Killing form on the dual of the Cartan subalgebra of L is completely determined by the Cartan matrix; namely

κ(α_{i}, α_{j}) = \frac{4t_{ij}}{t_{ii}t_{jj}} , where
t_{ij} = ∑_{α∈Φ}⟨ α, α_{i} ⟩⟨ α, α_{j} ⟩.

Step 1. α + β is not a root.

In this case c_{α,β} = 0 and from now on we may suppose that α + β is a root. Step 2. height(α) > height(β).

Then c_{α,β} = −c_{β,α} and now we may suppose that height(α) ≤ height(β). Step 3. α is negative.

Step 3a. β is negative.

Then c_{α,β} = c_{−β,−α}. Step 3b. β is positive.

If ξ = α + β is negative, then c_{α,β} = \frac{κ(ξ, ξ)c_{β,−ξ}}{κ(α, α)} , whereas if ξ is positive,

then c_{α,β} = \frac{κ(ξ, ξ)c_{−α,ξ}}{κ(β, β)} . Return to Step 2.

Step 4. α is positive.

Let (ε, η) be the extraspecial pair for α + β so that α + β = ε + η. If ε = α, then c_{α,β} = r + 1, where r is the greatest integer such that β − rα is a root. If ε = β, then c_{α,β} = −1. Otherwise,

c_{α,β} = \frac{κ(α + β, α + β)}{r + 1} (\frac{c_{−ε,β}c_{−η,α}}{κ(β − ε, β − ε)} − \frac{c_{−ε,α}c_{−η,β}}{κ(α − ε, α − ε)}) ,

where r is the greatest integer such that η − rε is a root.

The same choice of structure constants is made in Gilkey and Seitz (1988).

For the case in which all elements of Φ have the same length a considerable improvement to the algorithm can be obtained via the following theorem. Furthermore, in this case, ad(e_{−α})e_{α+β} = c_{−α,α+β}e_{β} = c_{α,β}e_{β} (see Step 3b above) and so the matrix of ad(e_{−α})|_{V} is the transpose of the matrix of ad(e_{α})|_{V}.

Theorem 2. Given that Φ is a root system with fundamental roots Δ where Φ^{+} is totally ordered by ≺ (as above), suppose that all roots have the same length and that the structure constants are determined by the algorithm just described. Then for α ∈ Δ and β ∈ Φ such that α + β ∈ Φ where the expression for β as a sum of fundamental roots involves some ω ≻ α, we have

c_{α,β} = \{ \begin{matrix}1 & if β ∈ Φ^{+} \\ −1 & if β ∈ Φ^{−}. \end{matrix}

Proof. Suppose at first that β ∈ Φ^{+} so that α + β ∈ Φ^{+}. All roots have the same length and so the α-chain through β is β, α + β. By Carter (1972), §4.2 this forces c_{α,β} to be ±1.

Because α is positive, we determine c_{α,β} via Step 4 of the algorithm. If (ε, η) is the extraspecial pair for α + β, then α + β = ε + η and η − ε is not a root. Because ε ⪯ α, it is not possible to have ε = β whereas if ε = α, then c_{α,β} = 1 and in this case we have finished. Thus we may suppose that ε ≺ α. Then c_{α,β} = c_{−ε,β}c_{−η,α} since α − ε is not a root and therefore c_{−ε,α} = 0. Now c_{α,β} ≠ 0 and therefore c_{−ε,β} ≠ 0, whence δ = β − ε = η − α is a (positive) root which involves ω ≻ ε. Thus Step 3b of the algorithm shows that c_{−ε,β} = c_{ε,δ} = 1 by induction. Again using Step 3b we have c_{−η,α} = c_{α,δ} = 1 by induction. It follows that c_{α,β} = 1.

It remains to consider the case in which β ∈ Φ^{−} and α + β ∈ Φ. Then −α − β ∈ Φ^{+} and, as in Step 3 of the algorithm, we have c_{α,β} = −c_{α,−α−β} and the result follows from the case already dealt with.

Corollary 1. Suppose that Φ, Φ_{1}, Δ, Δ_{1} and X are as above with Δ\Δ_{1} = {ω}. If all roots have the same length, then c_{α,β} = 1 for all α ∈ Δ_{1}, β ∈ X such that α + β ∈ X.

3. Lie algebras and groups of types E_{6}, E_{7}, E_{8}, F_{4} and G_{2}

The Weyl character formula shows that the smallest degree of a non-trivial (irreducible) representation of the complex Lie algebra of type G_{2}, F_{4}, E_{6}, E_{7} and E_{8} is respectively 7, 26, 27, 56 and 248 (Tits (1967), pp. 41–50). Theorem 1 enables us to write down representations of the Lie algebras of types E_{6} and E_{7} of degrees 27 and 56 respectively whereas the dimension of the adjoint representation of E_{8} is 248.

In order to obtain representations of minimal degree for the Lie algebras of types G_{2} and F_{4} we realize them as the fixed points of graph automorphisms of D_{4} and E_{6} respectively (see § 3.4). The modules for the Lie algebras of types D_{4} and E_{6} obtained from Theorem 1 via the inclusions D_{4} ⊆ D_{5} and E_{6} ⊆ E_{7} restrict to modules for G_{2} and F_{4} of dimensions 8 and 27 respectively. In both cases there is an invariant submodule of codimension 1 affording representations of dimensions 7 and 26 respectively. It turns out that all the representations considered so far are subrepresentations of the adjoint representation of E_{8}.

The use of a Chevalley basis in our construction of the modules means that each module comes equipped with a ℤ-form, and therefore the usual Chevalley construction (Carter 1972, Chapter 4) yields matrix groups over all fields. In general the group so constructed is a central extension of the simple group. Steinberg (1962) has given pairs of generators for the simple groups, and hence we obtain pairs of generators for the central extensions.

In most cases the representations obtained are irreducible. However, when the Lie algebra has type G_{2} or F_{4} and the field has characteristic 2 or 3 respectively, there is a submodule of dimension 1 whose quotient is an irreducible module of dimension 6 or 25, respectively.

For the remainder of the paper we shall consider a simple Lie algebra L and a representation ϕ : L → 𝔤l(ℓ, ℂ) of L by ℓ × ℓ matrices. For each e_{α} ∈ L_{α} the matrix ϕ(e_{α}) will have integer entries and be the restriction of ad(e_{α}) to a suitable module, hence nilpotent. Moreover, for any field 𝔽 and ξ ∈ 𝔽 the sum

x_{α}(ξ) = exp(ξϕ(e_{α})) = 1 + ξϕ(e_{α}) + \frac{ξ^{2}}{2!} ϕ(e_{α})^{2} + ⋯

will be finite (it will never have more than 3 terms) and defined over 𝔽. The Chevalley group associated with L, ϕ and the field 𝔽_{q} of order q is

G_{ϕ}(q) = ⟨ x_{α}(ξ) | ξ ∈ 𝔽_{q}, α ∈ Φ ⟩.

In what follows we let I denote the identity matrix and E_{i,j} denote the elementary matrix with 1 in the ij-th position and 0’s elsewhere; ξ will be always be an element of the relevant field.

Following Carter (1972), §6.4, for each positive root α we take

n_{α}(ξ) = x_{α}(ξ)x_{−α}(−ξ^{−1})x_{α}(ξ) and
h_{α,ξ} = n_{α}(ξ)n_{α}(−1).

Then h_{α,ξ} satisfies

h_{α,ξ}x_{β}(ν)h_{α,ξ}^{−1} = x_{β}(ξ^{⟨ β,α ⟩}ν)

for all ν ∈ 𝔽_{q}^{×} and β ∈ Φ and so may be identified with the element h in Theorem 3.4 of Steinberg (1962). In each case the matrix h_{α,ξ} will be diagonal.

Finally, n will always be a product (in some order) of the n_{α}(1) where α ∈ Δ; that is, it will correspond to a Coxeter element of the Weyl group.

3.1. E_{8}. Label the fundamental roots as indicated in the diagram

• • • • • • • •_{a}  _{c}  ^{b}_{d}  _{e}  _{f}  _{g}  _{h}

The degree of the smallest representation is 248. This is the dimension of the Lie algebra of type E_{8}, and so the Chevalley construction for the field 𝔽_{q} produces the (simple) adjoint group E_{8}(q).

In this case, for e ∈ L, ϕ(e) is the matrix of ad(e) with respect to the Chevalley basis. The Lie algebra is generated by the matrices ϕ(e_{α}), where ±α ∈ {a, b, …, h}. Each matrix has at most 61 of its 61, 504 entries non-zero but this is still too large to be given here. However, Theorem 2 and the fact that ϕ(e_{−α}) is almost the transpose of ϕ(e_{α}) allows these matrices to be computed rapidly. From these we get generators for the group E_{8}(q):

x_{α}(ξ) = I + ξϕ(e_{α}) + \frac{1}{2}ξ^{2}ϕ(e_{α})^{2} for ± α ∈ \{a, b, …, h\},

where ξ ∈ 𝔽_{q}. Using these elements we have

n_{α}(ξ) = x_{α}(ξ)x_{−α}(−ξ^{−1})x_{α}(ξ) for α ∈ {a, b, …, h},
n = n_{h}(1)n_{g}(1)n_{f}(1)n_{e}(1)n_{d}(1)n_{c}(1)n_{b}(1)n_{a}(1) and
h_{h,ξ} = n_{h}(ξ)n_{h}(−1).

It follows from Steinberg (1962), Theorem 3.11 that E_{8}(q) is generated by

\{ \begin{matrix}h_{h,μ} and x_{h}(1)n & for q > 3 \\ x_{h}(1) and n & for q ≤ 3, \end{matrix}

where μ is a primitive element of 𝔽_{q}.

3.2. E_{7}. The Dynkin diagram for E_{7} is

• • • • • • •_{a}  _{c}  ^{b}_{d}  _{e}  _{f}  _{g}

labelled to be consistent with the diagram of type E_{8}. We use Theorem 1 with Δ_{1} = Δ(E_{7}), Δ = Δ(E_{8}) and γ = h to construct a representation space V of dimension 56 of the Lie algebra of type E_{7}. The basis for V is { e_{α} | α ∈ X } with X defined and ordered as in §2 (so that X = {2a+3b+4c+6d+5e+4f +3g+h, …, f +g+h, g+h, h}). The ordering ensures that the matrix ϕ(e_{α}) of ad(e_{α})|_{V} is upper triangular for each root α ∈ Φ^{+}(E_{7}). By Corollary 1, for all α ∈ Δ_{1}, every nonzero entry of ϕ(e_{α}) is 1.

Since ϕ(e_{−α}) is the transpose of ϕ(e_{α}) for any root α, the following elements and their transposes generate the Lie algebra:

ϕ(e_{a}) = E_{7,8} + E_{9,10} + E_{11,12} + E_{13,15} + E_{16,18} + E_{19,22} + E_{35,38}
+ E_{39,41} + E_{42,44} + E_{45,46} + E_{47,48} + E_{49,50},
ϕ(e_{b}) = E_{5,6} + E_{7,9} + E_{8,10} + E_{20,23} + E_{24,26} + E_{27,29} + E_{28,30}
+ E_{31,33} + E_{34,37} + E_{47,49} + E_{48,50} + E_{51,52},
ϕ(e_{c}) = E_{5,7} + E_{6,9} + E_{12,14} + E_{15,17} + E_{18,21} + E_{22,25} + E_{32,35}
+ E_{36,39} + E_{40,42} + E_{43,45} + E_{48,51} + E_{50,52},
ϕ(e_{d}) = E_{4,5} + E_{9,11} + E_{10,12} + E_{17,20} + E_{21,24} + E_{25,28} + E_{29,32}
+ E_{33,36} + E_{37,40} + E_{45,47} + E_{46,48} + E_{52,53},
ϕ(e_{e}) = E_{3,4} + E_{11,13} + E_{12,15} + E_{14,17} + E_{24,27} + E_{26,29} + E_{28,31}
+ E_{30,33} + E_{40,43} + E_{42,45} + E_{44,46} + E_{53,54},
ϕ(e_{f}) = E_{2,3} + E_{13,16} + E_{15,18} + E_{17,21} + E_{20,24} + E_{23,26} + E_{31,34}
+ E_{33,37} + E_{36,40} + E_{39,42} + E_{41,44} + E_{54,55} and
ϕ(e_{g}) = E_{1,2} + E_{16,19} + E_{18,22} + E_{21,25} + E_{24,28} + E_{26,30} + E_{27,31}
+ E_{29,33} + E_{32,36} + E_{35,39} + E_{38,41} + E_{55,56}.

The matrix group G_{ϕ}(q) is generated by the elements x_{α}(ξ) = I +ξϕ(e_{α}) (as ϕ(e_{α})^{2} = 0) for ±α ∈ {a, b, …, g} and ξ ∈ 𝔽_{q}.

In general we have n_{α}(ξ) = x_{α}(ξ)x_{−α}(−ξ^{−1})x_{α}(ξ) and h_{α,ξ} = n_{α}(ξ)n_{α}(−1). The centre of G_{ϕ}(q) is generated by h_{b,−1}h_{e,−1}h_{g,−1} = −I and thus G_{ϕ}(q) is 2E_{7}(q) (the central extension of the simple group E_{7}(q) by a group of order 2) when q is odd or E_{7}(q) when q is a power of 2.

The diagonal matrix h_{g,ξ} has its ith diagonal entry equal to

ξ if i = 1, 16, 18, 21, 24, 26, 27, 29, 32, 35, 38 or 55,
ξ^{−1} if i = 2, 19, 22, 25, 28, 30, 31, 33, 36, 39, 41 or 56,
1 otherwise.

A Coxeter element is the matrix n = n_{g}(1)n_{f}(1)n_{e}(1)n_{d}(1)n_{c}(1)n_{b}(1)n_{a}(1). Its nonzero entries are equal to 1 for the indices

(1, 10), (9, 12), (10, 14), (11, 15), (12, 17), (13, 18), (14, 23), (15, 21), (16, 22),
(17, 26), (18, 25), (19, 5), (20, 29), (21, 30), (22, 7), (23, 38), (24, 33), (26, 41),
(27, 37), (29, 44), (32, 46), (35, 50), (38, 52), (39, 32), (41, 35), (42, 36), (44, 39),
(45, 40), (46, 42), (47, 43), (48, 45), (51, 49), (52, 53), (53, 54), (54, 55), (55, 56)

and −1 for

\begin{matrix}(2, 1), (3, 2), (4, 3), (5, 4), (6, 8), (7, 6), (8, 9), (25, 11), (28, 13), (30, 20), (31, 16), \\ (33, 24), (34, 19), (36, 27), (37, 28), (40, 31), (43, 34), (49, 48), (50, 51), (56, 47). \end{matrix}

It follows from Steinberg (1962), Theorem 3.11 that G_{ϕ}(q) is generated by

\{ \begin{matrix}h_{g,μ} and x_{g}(1)n & for q > 3 \\ x_{g}(1) and n & for q ≤ 3, \end{matrix}

where μ is a primitive element of 𝔽_{q}.

3.3. E_{6}. The Dynkin diagram for E_{6} is

• • • • • •_{a}  _{c}  ^{b}_{d}  _{e}  _{f}

labelled to be consistent with the diagram of type E_{7}. We use Theorem 1 with Δ_{1} = Δ(E_{6}), Δ = Δ(E_{7}) and γ = g to construct a representation space V of dimension 27 of the Lie algebra of type E_{6}. The basis of V is indexed by X and ordered as in §2, that is, X = {2a + 2b + 3c + 4d + 3e + 2f + g, …, e + f + g, f + g, g}. This choice of ordering means that for α ∈ Δ_{1} the matrix ϕ(e_{α}) of ad(e_{α})|_{V} is upper triangular. (Note that there are two 27-dimensional representations of the Lie algebra of type E_{6}; one is the contragredient of the other.) By Corollary 1 the nonzero entries of the matrices are 1 and a straightforward calculation produces the following 27 × 27 matrices:

ϕ(e_{a}) = E_{1,2} + E_{11,13} + E_{14,16} + E_{17,18} + E_{19,20} + E_{21,22},
ϕ(e_{b}) = E_{4,5} + E_{6,7} + E_{8,10} + E_{19,21} + E_{20,22} + E_{23,24},
ϕ(e_{c}) = E_{2,3} + E_{9,11} + E_{12,14} + E_{15,17} + E_{20,23} + E_{22,24},
ϕ(e_{d}) = E_{3,4} + E_{7,9} + E_{10,12} + E_{17,19} + E_{18,20} + E_{24,25},
ϕ(e_{e}) = E_{4,6} + E_{5,7} + E_{12,15} + E_{14,17} + E_{16,18} + E_{25,26} and
ϕ(e_{f}) = E_{6,8} + E_{7,10} + E_{9,12} + E_{11,14} + E_{13,16} + E_{26,27}.

The matrix ϕ(e_{−α}) is the transpose of ϕ(e_{α}) and the Lie algebra of type E_{6} is generated by the ϕ(e_{α}) for ±α ∈ {a, …, f}.

The matrix group G_{ϕ}(q) is generated by the elements x_{α}(ξ) = exp(ξϕ(e_{α})) = I + ξϕ(e_{α}) for ±α ∈ {a, b, …, f} and ξ ∈ 𝔽_{q}.

If q ≡ 1 mod 3, the field 𝔽_{q} contains an element ω of order 3 and the centre of G_{ϕ}(q) is generated by the scalar matrix h_{a,ω^{2}}h_{c,ω}h_{e,ω^{2}}h_{f,ω} of order 3. Thus when q ≡ 1 mod 3, the group G_{ϕ}(q) is 3E_{6}(q) (the central extension of the simple group E_{6}(q) by a group of order 3); otherwise it is the simple group E_{6}(q).

The diagonal matrix h_{a,ξ} has its ith diagonal entry equal to

ξ if i = 1, 11, 14, 17, 19 or 21
ξ^{−1} if i = 2, 13, 16, 18, 20 or 22
1 otherwise.

A Coxeter element is given by n = n_{a}(1)n_{c}(1)n_{d}(1)n_{e}(1)n_{f}(1)n_{b}(1); it has nonzero entries equal to 1 for indices

\begin{matrix}(1, 10), (7, 12), (9, 14), (10, 15), (11, 16), (12, 17), (14, 18), (15, 21), (17, 22), \\ (18, 9), (19, 24), (20, 11), (21, 27), (22, 19), (23, 13), (24, 20), (25, 23) \end{matrix}

and −1 for

(2, 1), (3, 2), (4, 3), (5, 8), (6, 5), (8, 7), (13, 4), (16, 6), (26, 25), (27, 26).

It follows from Steinberg (1962), Theorem 3.11 that G_{ϕ}(q) is generated by

\{ \begin{matrix}h_{a,μ} and x_{a}(1)n & for q > 3 \\ x_{a}(1) and n & for q = 2 or 3, \end{matrix}

where μ is a primitive element of 𝔽_{q}.

3.4. The fixed points of a graph automorphism. Suppose that all the roots of the simple Lie algebra L have the same length and that σ is an automorphism of the Dynkin diagram of L. Then σ determines a linear transformation of the Cartan subalgebra (and its dual) which permutes the fundamental roots Δ and fixes the full root system Φ. In fact this permutation of Φ, which we also denote by σ, extends to an automorphism of L such that, in the notation of §2, h_{α}^{σ} = h_{α^{σ}} and e_{α}^{σ} = ±e_{α^{σ}} (Carter 1972, Proposition 12.2.3).

Let L^{σ} be the subalgebra of fixed points of σ. In the cases of interest to us, namely D_{4} and E_{6}, L^{σ} is a simple Lie algebra of type G_{2} or F_{4}, respectively.

For each e ∈ L let ẽ be the sum of the elements in the ⟨ σ ⟩-orbit of e. Then for h ∈ H such that h^{σ} = h and for α ∈ Φ we have [h ẽ_{α}] = α(h)ẽ_{α}. Thus the roots of L^{σ} are the restrictions to H^{σ} = H ∩ L^{σ} of the roots of L. Also, for α ∈ Δ we have h̃_{α} = [ẽ_{α} ẽ_{−α}].

For types D_{4} and E_{6} and α ∈ Δ such that α ≠ α^{σ} we have ⟨ α, α^{σ} ⟩ = 0. Thus α(h̃_{α}) = 2 and it follows that a Chevalley basis for L^{σ} can be obtained by taking elements ẽ, where e runs through a set of representatives for the orbits of the group ⟨ −1, σ ⟩ on a Chevalley basis of L. We label the fundamental roots of L^{σ} as A, B, . . .. If A is the restriction of the root a, we choose e_{A} = ẽ_{a}, and so on. The remaining elements of the Chevalley basis are determined by e_{A}, e_{B}, … up to a sign and we choose the signs to agree with the algorithm of §2. The Cartan matrix of L^{σ} can be obtained from the Cartan matrix of L by taking, for each pair of representatives of the ⟨ σ ⟩-orbits on Δ, the sum of the entries in the row of α corresponding to the columns indexed by the elements in the ⟨ σ ⟩-orbit of β.

3.5. F_{4}. We begin with the complex Lie algebra L of type E_{6} given in §3.3 and with its Dynkin diagram as given there. In this case the graph automorphism has order 2. The restrictions of a, c, d and b to H^{σ} are the fundamental roots A, B, C and D of L^{σ}, and we have e_{A} = ẽ_{a} = e_{a} + e_{f}, e_{B} = ẽ_{c} = e_{c} + e_{e}, e_{C} = ẽ_{d} = e_{d} and e_{D} = ẽ_{b} = e_{b}.

Since ( \begin{matrix}2 & 0 & −1 & 0 & 0 & 0 \\ 0 & 2 & 0 & −1 & 0 & 0 \\ −1 & 0 & 2 & −1 & 0 & 0 \\ 0 & −1 & −1 & 2 & −1 & 0 \\ 0 & 0 & 0 & −1 & 2 & −1 \\ 0 & 0 & 0 & 0 & −1 & 2 \end{matrix} ) is the Cartan matrix of L, the Cartan matrix

of L^{σ} with respect to the ordering A, B, C, D is ( \begin{matrix}2 & −1 & 0 & 0 \\ −1 & 2 & −1 & 0 \\ 0 & −2 & 2 & −1 \\ 0 & 0 & −1 & 2 \end{matrix} ) and therefore

its Dynkin diagram is

• • • •^{<}_{A}  _{B}  _{C}  _{D}

of type F_{4}.

Take ϕ to be the representation of dimension 27 for E_{6} with representation space V defined in §3.3 and let v_{1}, v_{2}, …, v_{27} be its basis defined there. We note that ϕ(e_{−X}) is the transpose of ϕ(e_{X}).

The restriction of ϕ to L^{σ} is reducible. There is an L^{σ}-invariant subspace U with basis u_{i}, i = 1, …, 26, where

u_{i} = \{ \begin{matrix}v_{i} & for 1 ≤ i < 13 \\ v_{13} + v_{14} & for i = 13 \\ v_{14} + v_{15} & for i = 14 \\ v_{i+1} & for 14 < i ≤ 26 \end{matrix}

In characteristic 0 the representation splits into the sum of the (irreducible) 26dimensional representation on U and the zero representation on ⟨u_{0}⟩, where u_{0} = v_{13} − v_{14} + v_{15}. In any characteristic ⟨u_{0}⟩ is the unique 1-dimensional L^{σ}-submodule of U.

If we let ψ denote the representation of L^{σ} on U we obtain the following matrix generators for the Lie algebra of type F_{4}.

ψ(e_{A}) = E_{1,2} + E_{6,8} + E_{7,10} + E_{9,12} + 2E_{11,13} + E_{11,14} + E_{13,15}
+ E_{16,17} + E_{18,19} + E_{20,21} + E_{25,26},
ψ(e_{B}) = E_{2,3} + E_{4,6} + E_{5,7} + E_{9,11} + E_{12,13} + 2E_{12,14} + E_{14,16}
+ E_{15,17} + E_{19,22} + E_{21,23} + E_{24,25},
ψ(e_{C}) = E_{3,4} + E_{7,9} + E_{10,12} + E_{16,18} + E_{17,19} + E_{23,24},
ψ(e_{D}) = E_{4,5} + E_{6,7} + E_{8,10} + E_{18,20} + E_{19,21} + E_{22,23},
ψ(e_{−A}) = E_{2,1} + E_{8,6} + E_{10,7} + E_{12,9} + E_{13,11} + 2E_{15,13} + E_{15,14}
+ E_{17,16} + E_{19,18} + E_{21,20} + E_{26,25},
ψ(e_{−B}) = E_{3,2} + E_{6,4} + E_{7,5} + E_{11,9} + E_{14,12} + E_{16,13} + 2E_{16,14}
+ E_{17,15} + E_{22,19} + E_{23,21} + E_{25,24},
ψ(e_{−C}) = ψ(e_{C})^{t} and
ψ(e_{−D}) = ψ(e_{D})^{t}.

From these we obtain

ψ(e_{2B+C}) = −E_{2,6} − E_{5,11} + E_{10,16} + E_{12,18} − E_{15,22} − E_{21,25} and
ψ(e_{−(2B+C)}) = ψ(e_{2B+C})^{t}.

The group G_{ψ}(q) is generated by the elements x_{α}(ξ) = I + ξψ(e_{α}) + \frac{1}{2}ξ^{2}ψ(e_{α})^{2} for ±α ∈ {A, B, C, D} and ξ ∈ 𝔽_{q}. It is isomorphic to the simple group F_{4}(q).

In particular we have:

x_{A}(1) = I + ψ(e_{A}) + \frac{1}{2}ψ(e_{A})^{2} = I + E_{1,2} + E_{6,8} + E_{7,10} + E_{9,12} + 2E_{11,13}
+ E_{11,14} + E_{13,15} + E_{16,17} + E_{18,19} + E_{20,21} + E_{25,26} + E_{11,15},
x_{C}(1) = I + ψ(e_{C}) = I + E_{3,4} + E_{7,9} + E_{10,12} + E_{16,18} + E_{17,19} + E_{23,24} and
x_{−B}(1) = I + ψ(e_{−B}) + \frac{1}{2}ψ(e_{−B})^{2} = I + E_{3,2} + E_{6,4} + E_{7,5} + E_{11,9} + E_{14,12}
+ E_{16,12} + E_{16,13} + 2E_{16,14} + E_{17,15} + E_{22,19} + E_{23,21} + E_{25,24}.

The ith diagonal entry of h_{A,ξ} = n_{A}(ξ)n_{A}(−1) is equal to

ξ if i = 1, 6, 7, 9, 16, 18, 20 or 25
ξ^{−1} if i = 2, 8, 10, 12, 17, 19, 21 or 26
ξ^{2} if i = 11
ξ^{−2} if i = 15
1 otherwise

and the ith diagonal entry of h_{2B+C,ξ} is equal to

ξ if i = 2, 5, 10, 12, 15 or 21
ξ^{−1} if i = 6, 11, 16, 18, 22 or 25
1 otherwise.

The Coxeter element n = n_{A}(1)n_{B}(1)n_{C}(1)n_{D}(1) has nonzero entries equal to 1 for indices

\begin{matrix}(1, 5), (4, 7), (5, 9), (6, 10), (7, 12), (9, 20), (11, 21), (13, 14), (18, 23), (19, 16), \\ (20, 24), (21, 18), (22, 17), (23, 19), (24, 25), (25, 26), (26, 22) \end{matrix}

and −1 for

(2, 1), (3, 2), (8, 3), (10, 4), (12, 11), (14, 13), (14, 14), (15, 6), (16, 15), (17, 8).

It follows from Steinberg (1962), Theorems 3.11 and 3.14 that F_{4}(q) is generated by

h_{A,μ} and x_{A}(1)n  for q odd and q > 3
x_{A}(1) and n  for q = 3
h_{2B+C,μ} and x_{C}(1)x_{−B}(1)n for q even and q > 2
x_{C}(1)x_{−B}(1) and n  for q = 2,

where μ is a primitive element of 𝔽_{q}.

Except when 𝔽_{q} has characteristic 3 this representation of F_{4}(q) is irreducible. In characteristic 3 we have u_{0} = u_{13} + u_{14} ∈ U and the quotient U/⟨u_{0}⟩ affords an irreducible representation of F_{4}(q) of dimension 25.

3.6. G_{2}. Let a, b, c, d and e be the fundamental roots for the root system of type D_{5} as shown in the Dynkin diagram:

[figure: •; •; •; •; •; a; b; c; d; e]

The root system of type D_{5} has 20 positive roots and that of D_{4} has 12. The ordering given in §2 for the roots in Φ^{+}(D_{5}) \ Φ^{+}(D_{4}) is a + b + 2c + 2d + e, a + b + 2c + d + e, a + b + c + d + e, b + c + d + e, a + c + d + e, c + d + e, d + e, e. Applying the methods of §2 gives an 8-dimensional representation ϕ of the Lie algebra L of type D_{4} such that

ϕ(e_{a}) = E_{3,4} + E_{5,6},
ϕ(e_{b}) = E_{3,5} + E_{4,6},
ϕ(e_{c}) = E_{2,3} + E_{6,7},
ϕ(e_{d}) = E_{1,2} + E_{7,8}

and where ϕ(e_{−α}) is the transpose of ϕ(e_{α}) for all α ∈ Δ.

We regard the Lie algebra of type G_{2} as the subalgebra of fixed points L^{σ} of the diagram automorphism σ of order three of L. The restrictions of a and c to H^{σ} are the fundamental roots A and B for L^{σ} and we have e_{A} = ẽ_{a} = e_{a}+e_{b}+e_{d} and e_{B} = ẽ_{c} = e_{c}.

The Cartan matrix of L is ( \begin{matrix}2 & 0 & −1 & 0 \\ 0 & 2 & −1 & 0 \\ −1 & −1 & 2 & −1 \\ 0 & 0 & −1 & 2 \end{matrix} ) and the considerations of §3.4 show

that the Cartan matrix for L^{σ} is ( \begin{matrix}2 & −1 \\ −3 & 2 \end{matrix} ) and hence its Dynkin diagram is of

type G_{2}, namely [figure: <; A; B].

The restriction of ϕ to L^{σ} gives an 8-dimensional representation V of G_{2} with basis v_{1}, …, v_{8}. This representation is reducible and there is an L^{σ}-invariant submodule U with basis u_{i}, i = 1, …, 7, where

u_{i} = \{ \begin{matrix}v_{i} & for 1 ≤ i < 4 \\ v_{4} + v_{5} & for i = 4 \\ v_{i+1} & for 4 < i ≤ 7 \end{matrix}

In characteristic 0, we have V = U ⊕ ⟨u_{0}⟩ as L^{σ}-modules, where u_{0} = −v_{4} + v_{5}. In any characteristic, ⟨u_{0}⟩ is the unique 1-dimensional L^{σ}-submodule of V.

Let ψ denote the restriction of ϕ to L^{σ} with respect to the basis u_{i} of U. We have

ψ(e_{A}) = E_{1,2} + 2E_{3,4} + E_{4,5} + E_{6,7},
ψ(e_{−A}) = E_{2,1} + E_{4,3} + 2E_{5,4} + E_{7,6},
ψ(e_{B}) = E_{2,3} + E_{5,6},
ψ(e_{−B}) = E_{3,2} + E_{6,5}

and later we shall need

ψ(e_{A+B}) = E_{1,3} − 2E_{2,4} + E_{4,6} − E_{5,7},
ψ(e_{−A−B}) = −E_{3,1} + E_{4,2} − 2E_{6,4} + E_{7,5},
ψ(e_{2A+B}) = −2E_{1,4} + E_{2,5} + E_{3,6} − E_{4,7} and
ψ(e_{−2A−B}) = −E_{4,1} + E_{5,2} + E_{6,3} − 2E_{7,4}.

The group G_{ψ}(q) is generated by the elements x_{α}(ξ) = I + ξψ(e_{α}) + \frac{1}{2}ξ^{2}ψ(e_{α})^{2} for ±α ∈ {A, B} and ξ ∈ 𝔽_{q}; it is isomorphic to G_{2}(q).

In particular we have

x_{A}(ξ) = I + ξE_{1,2} + 2ξE_{3,4} + ξE_{4,5} + ξE_{6,7} + ξ^{2}E_{3,5},
x_{−A}(ξ) = I + ξE_{2,1} + ξE_{4,3} + 2ξE_{5,4} + ξE_{7,6} + ξ^{2}E_{5,3},
x_{B}(ξ) = I + ξe_{B} = I + ξE_{2,3} + ξE_{5,6},
x_{−B}(ξ) = I + ξe_{−B} = I + ξE_{3,2} + ξE_{6,5},
n = n_{A}(1)n_{B}(1) = E_{1,3} − E_{2,1} + E_{3,6} − E_{4,4} − E_{5,2} + E_{6,7} + E_{7,5},
h_{A,ξ} = ξE_{1,1} + ξ^{−1}E_{2,2} + ξ^{2}E_{3,3} + E_{4,4} + ξ^{−2}E_{5,5} + ξE_{6,6} + ξ^{−1}E_{7,7},
h_{B,ξ} = E_{1,1} + ξE_{2,2} + ξ^{−1}E_{3,3} + E_{4,4} + ξE_{5,5} + ξ^{−1}E_{6,6} + E_{7,7} and
h_{2A+B,ξ} = ξ^{2}E_{1,1} + ξE_{2,2} + ξE_{3,3} + E_{4,4} + ξ^{−1}E_{5,5} + ξ^{−1}E_{6,6} + ξ^{−2}E_{7,7}.

It follows from Steinberg (1962), Theorems 3.11 and 3.14 that G_{2}(q) is generated by

h_{2A+B,μ} and x_{B}(1)x_{−A}(1)n for q ≡ 0 mod 3
h_{A,μ} and x_{A}(1)n  for q ≢ 0 mod 3 and q ≠ 2
x_{A}(1) and n  for q = 2,

where μ is a primitive element of 𝔽_{q}.

Except for characteristic 2 this representation of G_{2}(q) is irreducible. In characteristic 2 the subspace spanned by the fourth basis element is fixed, and so by taking the quotient an irreducible representation of dimension 6 is obtained.

4. The twisted groups

This section deals with the twisted groups ^{2}B_{2}, ^{3}D_{4}, ^{2}E_{6}, ^{2}F_{4} and ^{2}G_{2}. Each twisted group ^{k}G(q) is generated by fixed points of an automorphism of G(q^{k}) or G(q) of order k which arises from a field automorphism and an automorphism of order k of the underlying graph of the Dynkin diagram. In the case of a single root length, the twisted group^{k}G(q) is a subgroup of G(q^{k}) and the field automorphism is the Frobenius map. On the other hand, when there is more than one root length, a power of the field automorphism is the Frobenius map over the prime field. A consequence of this is that kG(q) is defined only for some values of q, and is a subgroup of G(q). See Chapters 13 and 14 of Carter (1972) for the definition and a discussion of these groups.

For ξ ∈ 𝔽_{q} write ξ^{σ} for the image of ξ under the field automorphism, for α ∈ Φ write \overline{α} for the image of α under the map on Φ corresponding to the graph automorphism and write \overline{x} for the image of x under the associated group automorphism.

Except for ^{2}B_{2}(2), ^{2}F_{4}(2) and ^{2}G_{2}(3), the twisted groups are simple (Steinberg (1967), p. 186 and Carter (1972), Chapter 14).

4.1. The groups ^{2}B_{2}. The Dynkin diagrams of types B_{2} and C_{2} are the same ( • •^{>}_{a} _{b} ) and the underlying graph has an automorphism of order 2. When the order q of the field is an odd power of 2 there is a field automorphism which, when combined with the graph automorphism, induces an automorphism of the group B_{2}(q). The group 2B_{2}(q) of fixed points is also known as a Suzuki group. They were first described in Suzuki (1960) and then interpreted as twisted groups of Lie type by Ree (Tits 1960). Thus in this section 𝔽_{q} will denote the field of order q = 2^{2m+1} and σ will be the automorphism of 𝔽_{q} such that ξ^{σ} = ξ^{2^{m}}; that is, 2σ^{2} = 1. This field automorphism combined with the graph automorphism extends to an automorphism of the group

given by \overline{x_{X}(ξ)} = \{ \begin{matrix}x\overline{X}(ξ^{σ}) & X a long root \\ x\overline{X}(ξ^{2σ}) & X a short root \end{matrix} .

Even though we refer to the groups as ^{2}B_{2}(q), we use the four dimensional (symplectic) representation obtained from the embedding of C_{2} in C_{3}. The following elements of ^{2}B_{2}(q) will be needed:

x = x_{a}(1)x_{b}(1)x_{a+2b}(1) = ( \begin{matrix}1 & 1 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix} ),

n = ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) and

h(ξ) = h_{a}(ξ)\overline{h_{a}(ξ)} = h_{a}(ξ)h_{b}(ξ^{σ})
= diag(ξ^{σ}, ξ^{−σ+1}, ξ^{σ−1}, ξ^{−σ}).

It follows from Steinberg (1962), Theorem 4.1 that ^{2}B_{2}(q) is generated by

\{ \begin{matrix}h(μ) and xn & for q > 2 \\ x and n & for q = 2 \end{matrix}

where μ is a primitive element of 𝔽_{q}.

4.2. The groups ^{3}D_{4}. The simple group^{3}D_{4}(q) is a subgroup of the orthogonal group Ω^{+}(8, q^{3}), which we identify with the Chevalley group of type D_{4} over F_{q^{3}}. It is defined in terms of an automorphism of Ω^{+}(8, q^{3}) of order 3 which arises from the diagram automorphism of order 3 and the Frobenius map of order 3.

[figure: •; •; •; •; a; b; c; d]

For ξ ∈ 𝔽_{q^{3}} the Frobenius map sends ξ to ξ^{q}. The group automorphism is given by \overline{x_{α}(ξ)} = x\overline{α}(ξ^{q}).

We begin with the 8-dimensional representation of the Lie algebra of type D_{4} given in §3.6. From the matrices ϕ(e_{α}), ±α ∈ {a, b, c, d}, we obtain the following elements of the Chevalley group of type D_{4}.

x_{a}(ξ) = I + ξE_{3,4} + ξE_{5,6},
x_{b}(ξ) = I + ξE_{3,5} + ξE_{4,6},
x_{c}(ξ) = I + ξE_{2,3} + ξE_{6,7},
x_{d}(ξ) = I + ξE_{1,2} + ξE_{7,8}

and x_{−α}(ξ) = x_{α}(ξ)^{t} for α ∈ {a, b, c, d}. From these elements we derive

n_{a}(ξ) = E_{1,1} + E_{2,2} + ξE_{3,4} − ξ^{−1}E_{4,3} + ξE_{5,6} − ξ^{−1}E_{6,5} + E_{7,7} + E_{8,8},
n_{b}(ξ) = E_{1,1} + E_{2,2} + ξE_{3,5} + ξE_{4,6} − ξ^{−1}E_{5,3} − ξ^{−1}E_{6,4} + E_{7,7} + E_{8,8},
n_{c}(ξ) = E_{1,1} + ξE_{2,3} − ξ^{−1}E_{3,2} + E_{4,4} + E_{5,5} + ξE_{6,7} − ξ^{−1}E_{7,6} + E_{8,8},
n_{d}(ξ) = ξE_{1,2} − ξ^{−1}E_{2,1} + E_{3,3} + E_{4,4} + E_{5,5} + E_{6,6} + ξE_{7,8} − ξ^{−1}E_{8,7},
h_{a,ξ} = diag(1, 1, ξ, ξ^{−1}, ξ, ξ^{−1}, 1, 1),
h_{b,ξ} = diag(1, 1, ξ, ξ, ξ^{−1}, ξ^{−1}, 1, 1),
h_{c,ξ} = diag(1, ξ, ξ^{−1}, 1, 1, ξ, ξ^{−1}, 1) and
h_{d,ξ} = diag(ξ, ξ^{−1}, 1, 1, 1, 1, ξ, ξ^{−1}).

We take x_{R}(ξ) = x_{a}(ξ)x_{b}(ξ^{σ})x_{d}(ξ^{σ^{2}}) for ξ ∈ 𝔽_{q^{3}} and x_{c}(η) for η ∈ 𝔽_{q} to be the fundamental root elements for the group ^{3}D_{4}(q). Then we have

x_{R}(1) = I + E_{1,2} + E_{3,4} + E_{3,5} + E_{3,6} + E_{4,6} + E_{5,6} + E_{7,8},
n_{R}(1) = x_{R}(1)x_{−R}(−1)x_{R}(1) = n_{a}(1)n_{b}(1)n_{d}(1)
= E_{1,2} − E_{2,1} + E_{3,6} − E_{4,5} − E_{5,4} + E_{6,3} + E_{7,8} − E_{8,7},
n = n_{R}(1)n_{c}(1) = E_{1,3} − E_{2,1} + E_{3,7} − E_{4,5} − E_{5,4} − E_{6,2} + E_{7,8} + E_{8,6} and
h_{R}(ξ) = h_{a,ξ}h_{b,ξ^{σ}}h_{d,ξ^{σ^{2}}} = diag(ξ^{σ^{2}}, ξ^{−σ^{2}}, ξ^{σ+1}, ξ^{σ−1}, ξ^{−σ+1}, ξ^{−σ−1}, ξ^{σ^{2}}, ξ^{−σ^{2}}).

It follows from Steinberg (1962), Theorem 4.1 that ^{3}D_{4}(q) is generated by

h_{R}(μ) and x_{R}(1)n

where μ is a primitive element of the field 𝔽_{q^{3}}.

4.3. The groups ^{2}E_{6}. Label the fundamental roots as in §3.3:

• • • • • •_{a}  _{c}  ^{b}_{d}  _{e}  _{f}

The diagram automorphism of order 2, combined with the Frobenius map ξ ↦ ξ^{σ} = ξ^{q} of the field F_{q^{2}}, gives rise to an automorphism g ↦ \overline{g} of the group G_{ϕ}(q^{2}) of type E_{6} (defined in §3.3) such that \overline{x_{α}(ξ)} = x\overline{α}(ξ^{σ}). The fixed points of this automorphism form the group \overline{G}_{ϕ}(q) of type ^{2}E_{6}. If 3 divides q + 1 and if ω is an element of order 3 in 𝔽_{q^{2}}, then h_{a,ω^{2}}h_{f,ω}h_{c,ω}h_{e,ω^{2}} is an element of order 3 in the centre of \overline{G}_{ϕ}(q). In this case \overline{G}_{ϕ}(q) is the central extension 3^{2}E_{6}(q) of the twisted simple group ^{2}E_{6}(q); otherwise it is the simple group ^{2}E_{6}(q) itself.

As fundamental root elements for the group \overline{G}_{ϕ}(q) we take x_{R}(ξ) = x_{a}(ξ)x_{f}(ξ^{σ}) and x_{S}(ξ) = x_{c}(ξ)x_{e}(ξ^{σ}) for ξ ∈ 𝔽_{q^{2}} and x_{d}(η) and x_{b}(η) for η ∈ 𝔽_{q}. Then

x_{R}(1) = I + E_{1,2} + E_{6,8} + E_{7,10} + E_{9,12} + E_{11,13} + E_{11,14} + E_{11,16} + E_{13,16}
+ E_{14,16} + E_{17,18} + E_{19,20} + E_{21,22} + E_{26,27}, and
x_{S}(1) = I + E_{2,3} + E_{4,6} + E_{5,7} + E_{9,11} + E_{12,14} + E_{12,15} + E_{12,17} + E_{14,17}
+ E_{15,17} + E_{16,18} + E_{20,23} + E_{22,24} + E_{25,26}.

From these elements and similar expressions for x_{−R}(1) and x_{−S}(1) we derive n_{R}(1) = x_{R}(1)x_{−R}(−1)x_{R}(1) and n_{S}(1) = x_{S}(1)x_{−S}(−1)x_{S}(1).

Then a Coxeter element is given by n = n_{R}(1)n_{S}(1)n_{d}(1)n_{b}(1); it has nonzero entries equal to 1 for indices

\begin{matrix}(1, 5), (4, 7), (5, 9), (6, 10), (7, 12), (9, 21), (11, 22), (13, 15), (19, 24), (20, 17), \\ (21, 25), (22, 19), (23, 18), (24, 20), (25, 26), (26, 27), (27, 23) \end{matrix}

and −1 for

(2, 1), (3, 2), (8, 3), (10, 4), (12, 11), (14, 13), (15, 14), (16, 6), (17, 16), (18, 8).

The diagonal matrix h_{R}(ξ) = h_{a,ξ}h_{f,ξ^{σ}} has its ith diagonal entry equal to

ξ  i = 1, 17, 19 or 21
ξ^{−1} i = 2, 18, 20 or 22
ξ^{σ}  i = 6, 7, 9 or 26
ξ^{−σ} i = 8, 10, 12 or 27
ξ^{σ+1} i = 11
ξ^{σ−1} i = 13
ξ^{−σ+1} i = 14
ξ^{−σ−1} i = 16
1  otherwise.

By Steinberg (1962), Theorem 4.1 ^{2}E_{6}(q) is generated by h_{R}(μ) and x_{R}(1)n where μ is a primitive element of 𝔽_{q^{2}}.

4.4. The groups ^{2}F_{4} and the Tits group. The Dynkin diagram of type F_{4} is

• • • •^{<}_{A}  _{B}  _{C}  _{D}

The underlying graph has an automorphism of order 2. The groups ^{2}F_{4}(q) were introduced by Ree (1961a) and are defined only when the field has order an odd power of 2. Thus let 𝔽_{q} be the field of order q = 2^{2m+1}. There is an automorphism σ of 𝔽_{q} such that ξ^{σ} = ξ^{2^{m}}; that is, 2σ^{2} = 1. This field automorphism combined with the graph automorphism X ↦ \overline{X} extends to an automorphism of the group given by

\overline{x_{X}(ξ)} = \{ \begin{matrix}x\overline{X}(ξ^{σ}) & X a long root \\ x\overline{X}(ξ^{2σ}) & X a short root \end{matrix} .

We use the representation and elements given in §3.5. Let x = x_{B}(1)x_{C}(1)x_{2B+C}(1); its matrix has 1’s on the diagonal and the other nonzero entries are 1 for indices

(2, 3), (2, 4), (2, 6), (3, 4), (4, 6), (5, 7), (5, 9), (5, 11), (7, 9), (9, 11), (10, 12), (10, 16), (10, 18), (12, 13), (12, 16), (14, 16), (14, 18), (15, 17), (15, 19), (15, 22), (16, 18), (17, 19), (19, 22), (21, 23), (21, 24), (21, 25), (23, 24), (24, 25).

The matrix for

n = n_{2B+C}(1)n_{C}(1)n_{A}(1)n_{D}(1) = (n_{B}(1)n_{C}(1))^{3}n_{A}(1)n_{D}(1) =
x_{2B+C}(1)x_{C}(1)x_{−2B−C}(−1)x_{−C}(−1) ×
x_{2B+C}(1)x_{C}(1)x_{A}(1)x_{D}(1)x_{−A}(−1)x_{−D}(−1)x_{A}(1)x_{D}(1)

has nonzero entries equal to 1 for indices

(1, 2), (2, 10), (3, 5), (4, 3), (5, 15), (6, 1), (7, 12), (8, 7), (9, 8), (10, 21),
(11, 4), (12, 17), (13, 13), (13, 14), (14, 14), (15, 23), (16, 9), (17, 20), (18, 6),
(19, 16), (20, 19), (21, 26), (22, 11), (23, 24), (24, 22), (25, 18), (26, 25).

The diagonal matrix h(ξ) = h_{B,ξ}\overline{h_{B,ξ}} = h_{B,ξ}h_{C,ξ^{σ}} has ith diagonal entry

ξ  i = 2, 5, 15 or 21
ξ^{−1} i = 6, 11, 22 or 25
ξ^{−σ+1} i = 4, 9, 19 or 24
ξ^{σ−1} i = 3, 7, 17 or 23
ξ^{σ}  i = 10
ξ^{−σ+2} i = 12
ξ^{σ−2} i = 16
ξ^{−σ} i = 18
1  otherwise.

It follows from Steinberg (1962), Theorem 4.1 that ^{2}F_{4}(q) is generated by

\{ \begin{matrix}h(μ) and xn & for q > 2 \\ x and n & for q = 2 \end{matrix}

where μ is a primitive element of 𝔽_{q}.

The Tits group is the simple group (^{2}F_{4}(2))′; it has index 2 in ^{2}F_{4}(2) (Carter 1972, §14.4). The group (^{2}F_{4}(2))′ is generated by

xnx^{−1}n^{−1} and n.

4.5. The groups ^{2}G_{2}. Ree (1961b) describes these groups and gives a matrix representation for them equivalent to the one given here.

Let q = 3^{2m+1} and let σ be the automorphism of 𝔽_{q} defined by ξ^{σ} = ξ^{3^{m}} (and so 3σ^{2} = 1). The field automorphism σ combined with the graph automorphism X ↦ \overline{X} (interchanging roots A and B) extends to an automorphism of the group given by

\overline{x_{X}(ξ)} = \{ \begin{matrix}x\overline{X}(ξ^{σ}) & X a long root \\ x\overline{X}(ξ^{3σ}) & X a short root \end{matrix} .

We use the notation from §3.6, but choose different structure constants for the Lie algebra of type G_{2} to ensure that the graph automorphism has a nice form (Carter 1972, §12.4). The structure constants are determined by putting c_{α,β} = −(r + 1) instead of r + 1 whenever (α, β) is the extraspecial pair for α + β in the algorithm in §2. We obtain the following elements of ^{2}G_{2}(q).

x(ξ) = x_{A}(ξ^{σ})x_{B}(ξ)x_{A+B}(ξ^{σ+1})x_{2A+B}(ξ^{2σ+1}),

x(1) = ( \begin{matrix}1 & 1 & 0 & 0 & 1 & 2 & 2 \\ 0 & 1 & 1 & 2 & 1 & 0 & 1 \\ 0 & 0 & 1 & 2 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ),

n′ = (n_{A}(1)n_{B}(1))^{3} = E_{1,7} − E_{2,6} + E_{3,5} − E_{4,4} + E_{5,3} − E_{6,2} + E_{7,1} and
h(ξ) = h_{A}(ξ)\overline{h_{A}(ξ)} = h_{A}(ξ)h_{B}(ξ^{3σ})
= ξE_{1,1} + ξ^{3σ−1}E_{2,2} + ξ^{2−3σ}E_{3,3} + E_{4,4} + ξ^{3σ−2}E_{5,5} + ξ^{1−3σ}E_{6,6} + ξ^{−1}E_{7,7}.

It follows from Steinberg (1962), Theorem 4.1 that ^{2}G_{2}(q) is generated by h(μ) and x(1)n′ where μ is a primitive element of 𝔽_{q}.

5. Availability

The generators for all the groups described in this paper have been included in Magma since V2.4.

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   http://www.maths.usyd.edu.au:8000/u/don.

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Notes in Mathematics. Springer-Verlag, Berlin Heidelberg.

School of Mathematics and Statistics, University of Sydney, Australia

  E-mail address: R.Howlett@maths.usyd.edu.au

School of Science, University of Western Sydney, Nepean, Australia

  E-mail address: L.Rylands@nepean.uws.edu.au

School of Mathematics and Statistics, University of Sydney, Australia

  E-mail address: D.Taylor@maths.usyd.edu.au