A Catalogue of Classical Groups of Lie Type

Let V be a vector space over a field K with basis e_{1}, e_{2}, …, e_{n}.

1 The Linear Groups

Definition 1

GL_{n} (K) = {T ∈ M_{n} (K) : det (T) ≠ 0}
SL_{n} (K) = {T ∈ GL_{n} (K) : det (T) = 1}
PGL_{n} (K) = GL_{n} (K) /Z
PSL_{n} (K) = SL_{n} (K) /Z

where Z is the center of whatever group it makes sense to be center of.

We have shown that GL_{n} (K) has B, N-pair with B the subgroup of upper triangular matrices and N the set of monomial matrices. We will derive from these sets B, N-pairs for all the groups in Definition 1

Exercise 1 (Bourbaki) Let G be a finite group with Z ◁ G, B ≤ G, N ≤ G, and Z ≤ B. Let φ : G → G/Z be the canonical homomorphism g ↦ gZ. Then {G, B, N, W, S} is a Tits system, that is, G has B, N-pair with Weyl group W and generators S, if and only if {G′, B′, N′, W′, S′} is a Tits system where ♣′ denotes the image of ♣ under φ.

2 The Symplectic Groups

Definition 2 A bilinear form ⟨·, ·⟩ : V × V → 𝕂 is called skew-symmetric if ⟨x, y⟩ = − ⟨y, x⟩ for all x, y ∈ V.

Assume V has a non-degenerate skew-symmetric bilinear form ⟨·, ·⟩ and let A = (a_{i,j}) = (⟨e_{i}, e_{j}⟩). Then ⟨x, y⟩ = x^{t}Ay where matrices and vectors are used interchangeably with maps and forms. All symbols are to be interpreted in whatever way makes sense. Life is short. It can be shown that there is a basis of V for which

A = (\begin{matrix}0 & 1 \\ −1 & 0 \\ & & 0 & 1 \\ & & −1 & 0 \\ & & & & . & . & . \\ & & & & & & & 0 & 1 \\ & & & & & & & −1 & 0 \end{matrix})

Definition 3 The Symplectic group Sp_{n} (K) is defined

Sp_{n} (K) = {T ∈ GL_{n} (K) : T AT^{t} = A}

Thus, the Symplectic group is the collection of linear transformations T which preserve the form ⟨·, ·⟩. Thus,

T ∈ Sp_{n} (K)  iff  ⟨T (x) , T (y)⟩ = ⟨x, y⟩  iff x^{t}T^{t}AT y = x^{t}Ay

for all x, y ∈ V so that T^{t}AT = A.

Remark 1 It can be shown that det (T) = 1 for T ∈ Sp_{n} (K), so there is no group SSp_{n} (K).

Example 1 In dimension 2, we compute

(\begin{matrix}a & c \\ b & d \end{matrix}) (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) (\begin{matrix}a & b \\ c & d \end{matrix}) = (\begin{matrix}0 & ad − bc \\ bc − ad & 0 \end{matrix}) = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix})

iff ad − bc = 1 so that Sp_{n} (K) = SL (K).

Definition 4 Let Z denote the center of Sp_{n} (K). Then Z = {I, −I} and we define

PSp_{n} (K) = Sp_{n} (K) /Z.

3 The Orthogonal Groups

Definition 5 A bilinear form ⟨·, ·⟩ : V × V → 𝕂 is called symmetric if ⟨x, y⟩ = ⟨y, x⟩ for all x, y ∈ V.

Assume now that ⟨·, ·⟩ is a non-degenerate symmetric bilinear form. Assume also that K is finite and of characteristic ≠ 2. The orthogonal groups are defined differently for fields of characteristic 2. Again, let A = (⟨e_{i}, e_{j}⟩). Then it can be shown that if n = 2m + 1, there is a basis of V with either

A = (\begin{matrix}1 \\ & & I_{m} \\ & I_{m} \end{matrix})  or A = (\begin{matrix}ϵ \\ & & ϵI_{m} \\ & ϵI_{m} \end{matrix})  (1)

for ϵ ∈ K a non-square, and if n = 2m, there is a basis of V with either

A = (\begin{matrix}0 & I_{m} \\ I_{m} & 0 \end{matrix})  (2)

or A = (\begin{matrix}0 & I_{m−1} \\ I_{m−1} & 0 \\ & & 1 & 0 \\ & & 0 & −ϵ \end{matrix})  (3)

for ϵ ∈ K a non-square.

Definition 6 The orthogonal group

{T ∈ GL_{n} (K) : T^{t}AT = A}

is denoted

O_{2n+1} (K) if A is as in (1),
O_{2n}^{+} (K) if A is as in (2),
or  O_{2n}^{−} (K)  if A is as in (3)

Thus, the orthogonal groups are the collections of transformations T which preserve the form ⟨·, ·⟩.

Definition 7

SO_{n}^{♣} (K) = {T ∈ O_{n} (K) : det (T) = 1}
PO_{n}^{♣} (K) = O_{n}^{♣} (K) /Z
PSO_{n}^{♣} (K) = SO_{n}^{♣} (K) /Z

4 The Unitary Groups

Let λ → λ̄ be an automorphsim of K of order 2.

Definition 8 A map ⟨·, ·⟩ : V × V → 𝕂 is called a Hermitian scalar product if

⟨x_{1} + x_{2}, y⟩ = ⟨x_{1}, y⟩ + ⟨x_{2}, y⟩
⟨x, y_{1} + y_{2}⟩ = ⟨x, y_{1}⟩ + ⟨x, y_{2}⟩
⟨αx, y⟩ = α ⟨x, y⟩
⟨x, αy⟩ = ᾱ ⟨x, y⟩
⟨y, x⟩ = \overline{⟨x, y⟩}

Let ⟨, ·, ⟩ be a Hermitian scalar product. for all x, y ∈ V and all α ∈ K.

It can be shown that if K is finite, then K = 𝔽_{q^{2}} and the automorphism λ → λ̄ is given by λ → λ^{q} and that there is a basis of V for which A = (⟨e_{i}, e_{j}⟩) = I. Thus, ⟨u, v⟩ = \overline{u}^{t}v.

Definition 9 The unitary group U_{n} (K) is defined

U_{n} (K) = \{T ∈ GL_{n} (K) : \overline{T}^{t}T = I\} .

Thus, the unitary group is the collections of transformations T which preserve the form ⟨·, ·⟩.

Definition 10

SU_{n} (K) = {T ∈ U_{n} (K) : det (T) = 1}
PU_{n} (K) = U_{n} (K) /Z
PSU_{n} (K) = SU_{n} (K) /Z