March 1

Definition 1 A finite group with BN-pair is a finite group G with subgroups B, N ≤ G satisfying

1. G = ⟨B, N⟩

2. B ∩ N ◁ N

3. Let W = N/ (B ∩ N) and for each w ∈ W let ẇ ∈ N be a coset representative. Then W is generated by a set S = {s_{1}, s_{2}, …, s_{n}} of involutions such that

ṡ_{i}Bẇ ⊂ BẇB ∪ Bṡ_{i}wB  and  ṡ_{i}Bṡ_{i} ≠ B  (1)

for each w ∈ W and s_{i} ∈ S

We write T = B ∩ N. The group W = N/T is called the Weyl group associated with the BN-pair and n = |S| is called the rank of the BN-pair.

Example 1 (GL_{n} (K) is a group with BN-pair) Let K be any field and let

G = GL_{n+1} (K)
B = upper triangular matrices in G
N = monomial matrices in G
T = B ∩ N = diagonal matrices in G

where a monomial matrix is an element g ∈ G in which each row and each column contains exactly one nonzero entry. Then B and N form a BN-pair of rank n with W ≅ S_{n+1}.

Theorem 1 (Bruhat Decomposition) Let G be a finite group with BN-pair. Then

G = ˙⋃_{ẇ∈W} BẇB

Example 2 (Bruhat Decomposition of GL_{2} (𝔽_{2}))

G = \{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 1 \\ 1 & 0 \end{matrix}) , (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) , (\begin{matrix}0 & 1 \\ 1 & 1 \end{matrix}) , (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})\}
B = \{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})\}
N = \{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})\}
T = \{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix})\}

G = \{ \{(\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) , (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix})\} , \{(\begin{matrix}1 & 1 \\ 1 & 0 \end{matrix}) , (\begin{matrix}1 & 0 \\ 1 & 1 \end{matrix}) , (\begin{matrix}0 & 1 \\ 1 & 1 \end{matrix}) , (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix})\} \} = B ∪ B (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) B

Let G be a group with BN-pair B and N with generators S. Let W = N/ (B ∩ N) = N/T be the associated Weyl group.

Proposition 1 The set S ⊂ W satisfies the exchange condition. That is, if

ℓ (s_{i_{1}}s_{i_{2}} … s_{i_{m}}) = m
but  ℓ (s_{i_{0}}s_{i_{1}}s_{i_{2}} … s_{i_{m}}) < m + 1

then there exists 1 ≤ k ≤ m with

s_{i_{1}}s_{i_{2}} … s_{i_{k}} = s_{i_{0}}s_{i_{1}} … s_{i_{k−1}}.

Corollary 1 W is a Coxeter group

Definition 2 A finite group G is said to have a split BN-pair of rank n and characteristic p if the following conditions are satisfied.

1. G has a BN-pair of rank n such that

T = B ∩ N = ⋂_{x∈N} x^{−1}Bx,

2. There exists U ◁ B with B = U ⋊ T, and

3. U is the unique maximal normal p-subgroup of B and T is an Abelian p′-group, that is, p |̸ |T |.

Example 3 (GL_{n} (F_{q}) is a finite group with split BN-pair) Let F_{q} be a field of characteristic p. As before we let

G = GL_{n+1} (F_{q})
B = upper triangular matrices in G
N = monomial matrices in G
T = B ∩ N = diagonal matrices in G

In addition we let

U = upper triangular unit matrices in G

so an element of U is an upper triangular matrix with 1’s on the diagonal.

Then {G, B, U, N, T } is a finite group with split BN-pair of rank n and characteristic p with W ≅ S_{n+1}.