Demonstration

Let G = GL_{5} (K). Then G has BN pair with W = B/B ∩ N = S_{5} and generators S given by

(1)  s_{1} = (12)  s_{2} = (23)  s_{3} = (34)  s_{4} = (45)

so that

ṡ_{1} = (\begin{matrix}& 1 \\ 1 \\ & & 1 \\ & & & 1 \\ & & & & 1 \end{matrix}) ṡ_{2} = (\begin{matrix}1 \\ & & 1 \\ & 1 \\ & & & 1 \\ & & & & 1 \end{matrix}) ṡ_{3} = (\begin{matrix}1 \\ & 1 \\ & & & 1 \\ & & 1 \\ & & & & 1 \end{matrix}) ṡ_{4} = (\begin{matrix}1 \\ & 1 \\ & & 1 \\ & & & & 1 \\ & & & 1 \end{matrix}) .

Take

I = {(12) , (23) , (45)}
(2)  J = {(23) , (34) , (45)}
J′ = {(23) , (34)}

Using the CHEVIE package in GAP, (see last page) we compute the DDCRs of W_{I}\W/W_{J} to be 1 and (1432). Taking x = (1432), we have

K = I ∩ ^{x}J = {(12) , (23) , (45)} ∩ ^{x} {(23) , (34) , (45)} = {(12) , (23)}

so that W_{K} = ⟨(12) , (23)⟩. We will compute the Levi decomposition for P_{I}, P_{J}, P′_{J}, and P_{K} along with various other computations. Introduce a root system

Δ = {e_{i} − e_{j} : i ≠ j}

Δ_{+} = {e_{i} − e_{j} : i > j} (3)

on ℝ^{5} with fundamental system Π given by

(4)  α_{1} = e_{1} − e_{2} α_{2} = e_{2} − e_{3} α_{3} = e_{3} − e_{4} α_{4} = e_{4} − e_{5}.

Then

(5) Δ_{J} = ⟨Π_{J}⟩ ∩ Δ = ⟨α_{2}, α_{3}, α_{4}⟩ ∩ Δ = \{ \begin{matrix}& e_{2} − e_{3} & e_{2} − e_{4} & e_{2} − e_{5} \\ e_{3} − e_{2} & & e_{3} − e_{4} & e_{3} − e_{5} \\ e_{4} − e_{2} & e_{4} − e_{3} & & e_{4} − e_{5} \\ e_{5} − e_{2} & e_{5} − e_{3} & e_{5} − e_{4} \end{matrix} \}

and similarly for Δ_{J′} and Δ_{I}. Recall that for α = e_{i} − e_{j} we define

(6)  U_{α} = {E_{i,j} (t) : t ∈ ℝ} = {I + e_{i,j} (t) : t ∈ ℝ}

and

L_{J} = ⟨T, U_{α} : α ∈ Δ_{α}⟩
(7)  V_{J} = ⟨U_{α} : α ∈ Δ_{+} − Δ_{α}⟩

so that

(8) P_{J} = (\begin{matrix}♣ & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \end{matrix})  L_{J} = (\begin{matrix}♣ & 0 & 0 & 0 & 0 \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \end{matrix})  V_{J} = (\begin{matrix}1 & ♣ & ♣ & ♣ & ♣ \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

(9) P_{J′} = (\begin{matrix}♣ & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  L_{J′} = (\begin{matrix}♣ & 0 & 0 & 0 & 0 \\ 0 & ♣ & ♣ & ♣ & 0 \\ 0 & ♣ & ♣ & ♣ & 0 \\ 0 & ♣ & ♣ & ♣ & 0 \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  V_{J′} = (\begin{matrix}1 & ♣ & ♣ & ♣ & ♣ \\ 0 & 1 & 0 & 0 & ♣ \\ 0 & 0 & 1 & 0 & ♣ \\ 0 & 0 & 0 & 1 & ♣ \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

P_{J′} ∩ L_{J} = (\begin{matrix}♣ & 0 & 0 & 0 & 0 \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & ♣ & ♣ & ♣ & ♣ \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  L_{J′} = (\begin{matrix}♣ & 0 & 0 & 0 & 0 \\ 0 & ♣ & ♣ & ♣ & 0 \\ 0 & ♣ & ♣ & ♣ & 0 \\ 0 & ♣ & ♣ & ♣ & 0 \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  V_{J′} ∩ L_{J} = (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & ♣ \\ 0 & 0 & 1 & 0 & ♣ \\ 0 & 0 & 0 & 1 & ♣ \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) (10)

(11) P_{I} = (\begin{matrix}♣ & ♣ & ♣ & ♣ & ♣ \\ ♣ & ♣ & ♣ & ♣ & ♣ \\ ♣ & ♣ & ♣ & ♣ & ♣ \\ 0 & 0 & 0 & ♣ & ♣ \\ 0 & 0 & 0 & ♣ & ♣ \end{matrix})  L_{I} = (\begin{matrix}♣ & ♣ & ♣ & 0 & 0 \\ ♣ & ♣ & ♣ & 0 & 0 \\ ♣ & ♣ & ♣ & 0 & 0 \\ 0 & 0 & 0 & ♣ & ♣ \\ 0 & 0 & 0 & ♣ & ♣ \end{matrix})  V_{I} = (\begin{matrix}1 & 0 & 0 & ♣ & ♣ \\ 0 & 1 & 0 & ♣ & ♣ \\ 0 & 0 & 1 & ♣ & ♣ \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

(12) P_{K} = (\begin{matrix}♣ & ♣ & ♣ & ♣ & ♣ \\ ♣ & ♣ & ♣ & ♣ & ♣ \\ ♣ & ♣ & ♣ & ♣ & ♣ \\ 0 & 0 & 0 & ♣ & ♣ \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  L_{K} = (\begin{matrix}♣ & ♣ & ♣ & 0 & 0 \\ ♣ & ♣ & ♣ & 0 & 0 \\ ♣ & ♣ & ♣ & 0 & 0 \\ 0 & 0 & 0 & ♣ & 0 \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  V_{K} = (\begin{matrix}1 & 0 & 0 & ♣ & ♣ \\ 0 & 1 & 0 & ♣ & ♣ \\ 0 & 0 & 1 & ♣ & ♣ \\ 0 & 0 & 0 & 1 & ♣ \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

We compute the following conjugates. (13)

xP_{J} = (\begin{matrix}♣ & ♣ & ♣ & 0 & ♣ \\ ♣ & ♣ & ♣ & 0 & ♣ \\ ♣ & ♣ & ♣ & 0 & ♣ \\ ♣ & ♣ & ♣ & ♣ & ♣ \\ ♣ & ♣ & ♣ & 0 & ♣ \end{matrix})  ^{x}L_{J} = (\begin{matrix}♣ & ♣ & ♣ & 0 & ♣ \\ ♣ & ♣ & ♣ & 0 & ♣ \\ ♣ & ♣ & ♣ & 0 & ♣ \\ 0 & 0 & 0 & ♣ & 0 \\ ♣ & ♣ & ♣ & 0 & ♣ \end{matrix})  ^{x}V_{J} = (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ ♣ & ♣ & ♣ & 1 & ♣ \\ 0 & 0 & 0 & 0 & 1 \end{matrix})

P_{K} = (\begin{matrix}♣ & ♣ & ♣ & ♣ & ♣ \\ ♣ & ♣ & ♣ & ♣ & ♣ \\ ♣ & ♣ & ♣ & ♣ & ♣ \\ 0 & 0 & 0 & ♣ & ♣ \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  P_{I} ∩ ^{x} P_{J} = (\begin{matrix}♣ & ♣ & ♣ & 0 & ♣ \\ ♣ & ♣ & ♣ & 0 & ♣ \\ ♣ & ♣ & ♣ & 0 & ♣ \\ 0 & 0 & 0 & ♣ & ♣ \\ 0 & 0 & 0 & 0 & ♣ \end{matrix})  V_{I} = (\begin{matrix}1 & 0 & 0 & ♣ & ♣ \\ 0 & 1 & 0 & ♣ & ♣ \\ 0 & 0 & 1 & ♣ & ♣ \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) (14)

V_{K} = (\begin{matrix}1 & 0 & 0 & ♣ & ♣ \\ 0 & 1 & 0 & ♣ & ♣ \\ 0 & 0 & 1 & ♣ & ♣ \\ 0 & 0 & 0 & 1 & ♣ \\ 0 & 0 & 0 & 0 & 1 \end{matrix})  P_{I} ∩ ^{x}V_{J} = (\begin{matrix}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & ♣ \\ 0 & 0 & 0 & 0 & 1 \end{matrix})  V_{I} = (\begin{matrix}1 & 0 & 0 & ♣ & ♣ \\ 0 & 1 & 0 & ♣ & ♣ \\ 0 & 0 & 1 & ♣ & ♣ \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) (15)

gap> RequirePackage("chevie");
gap> W:=CoxeterGroup("A",4);
gap> WI:=Subgroup(W,[W.1,W.2,W.4]);
gap> WJ:=Subgroup(W,[W.2,W.3,W.4]);
gap> D:=DoubleCosets(W,WI,WJ);
gap> List(Elements(D[1]),i->CoxeterWord(W,i));
[ [  ], [ 4 ], [ 4, 3 ], [ 3 ], [ 3, 4, 3 ], [ 3, 4 ], [ 2, 1 ], [ 2, 1, 4 ],
[ 2, 1, 3, 2 ], [ 2, 1, 4, 3, 2 ], [ 2, 1, 3, 2, 4, 3 ],
[ 2, 1, 3, 2, 4, 3, 2 ], [ 4, 3, 2 ], [ 3, 2 ], [ 3, 4, 3, 2 ],
[ 3, 2, 4 ], [ 2 ], [ 2, 4 ], [ 2, 1, 3 ], [ 2, 1, 4, 3 ],
[ 2, 1, 3, 2, 4 ], [ 2, 1, 3, 4, 3, 2 ], [ 3, 2, 4, 3, 2 ], [ 3, 2, 4, 3 ],
[ 2, 4, 3, 2 ], [ 2, 3, 2 ], [ 2, 3 ], [ 2, 4, 3 ], [ 2, 1, 3, 4 ],
[ 2, 1, 3, 4, 3 ], [ 2, 3, 2, 4, 3, 2 ], [ 2, 3, 2, 4, 3 ],
[ 2, 3, 4, 3, 2 ], [ 2, 3, 2, 4 ], [ 2, 3, 4 ], [ 2, 3, 4, 3 ], [ 1 ],
[ 1, 4 ], [ 1, 4, 3 ], [ 1, 3 ], [ 1, 3, 4, 3 ], [ 1, 3, 4 ], [ 1, 2, 1 ],
[ 1, 2, 1, 4 ], [ 1, 2, 1, 3, 2 ], [ 1, 2, 1, 4, 3, 2 ],
[ 1, 2, 1, 3, 2, 4, 3 ], [ 1, 2, 1, 3, 2, 4, 3, 2 ], [ 1, 2 ], [ 1, 2, 4 ],
[ 1, 4, 3, 2 ], [ 1, 3, 2 ], [ 1, 3, 4, 3, 2 ], [ 1, 3, 2, 4 ],
[ 1, 2, 1, 3 ], [ 1, 2, 1, 4, 3 ], [ 1, 2, 1, 3, 2, 4 ],
[ 1, 2, 1, 3, 4, 3, 2 ], [ 1, 2, 3 ], [ 1, 2, 4, 3 ], [ 1, 2, 4, 3, 2 ],
[ 1, 2, 3, 2 ], [ 1, 3, 2, 4, 3, 2 ], [ 1, 3, 2, 4, 3 ], [ 1, 2, 1, 3, 4 ],
[ 1, 2, 1, 3, 4, 3 ], [ 1, 2, 3, 4 ], [ 1, 2, 3, 4, 3 ],
[ 1, 2, 3, 4, 3, 2 ], [ 1, 2, 3, 2, 4 ], [ 1, 2, 3, 2, 4, 3, 2 ],
[ 1, 2, 3, 2, 4, 3 ] ]
gap> List(Elements(D[2]),i->CoxeterWord(W,i));
[ [ 4, 3, 2, 1 ], [ 3, 2, 1 ], [ 3, 4, 3, 2, 1 ], [ 3, 2, 1, 4 ],
[ 3, 2, 1, 4, 3, 2, 1 ], [ 3, 2, 1, 4, 3, 2 ], [ 2, 1, 4, 3, 2, 1 ],
[ 2, 1, 3, 2, 1 ], [ 2, 1, 3, 2, 1, 4, 3, 2, 1 ],
[ 2, 1, 3, 2, 1, 4, 3, 2 ], [ 2, 1, 3, 2, 4, 3, 2, 1 ],
[ 2, 1, 3, 2, 1, 4, 3 ], [ 3, 2, 4, 3, 2, 1 ], [ 3, 2, 1, 4, 3 ],
[ 2, 4, 3, 2, 1 ], [ 2, 3, 2, 1 ], [ 2, 3, 2, 1, 4, 3, 2, 1 ],
[ 2, 3, 2, 1, 4, 3, 2 ], [ 2, 1, 3, 4, 3, 2, 1 ], [ 2, 1, 3, 2, 1, 4 ],
[ 2, 3, 2, 4, 3, 2, 1 ], [ 2, 3, 2, 1, 4, 3 ], [ 2, 3, 4, 3, 2, 1 ],
[ 2, 3, 2, 1, 4 ], [ 1, 4, 3, 2, 1 ], [ 1, 3, 2, 1 ], [ 1, 3, 4, 3, 2, 1 ],
[ 1, 3, 2, 1, 4 ], [ 1, 3, 2, 1, 4, 3, 2, 1 ], [ 1, 3, 2, 1, 4, 3, 2 ],
[ 1, 2, 1, 4, 3, 2, 1 ], [ 1, 2, 1, 3, 2, 1 ],
[ 1, 2, 1, 3, 2, 1, 4, 3, 2, 1 ], [ 1, 2, 1, 3, 2, 1, 4, 3, 2 ],
[ 1, 2, 1, 3, 2, 4, 3, 2, 1 ], [ 1, 2, 1, 3, 2, 1, 4, 3 ],
[ 1, 2, 4, 3, 2, 1 ], [ 1, 2, 3, 2, 1 ], [ 1, 3, 2, 4, 3, 2, 1 ],
[ 1, 3, 2, 1, 4, 3 ], [ 1, 2, 3, 2, 1, 4, 3, 2, 1 ],
[ 1, 2, 3, 2, 1, 4, 3, 2 ], [ 1, 2, 1, 3, 4, 3, 2, 1 ],
[ 1, 2, 1, 3, 2, 1, 4 ], [ 1, 2, 3, 4, 3, 2, 1 ], [ 1, 2, 3, 2, 1, 4 ],
[ 1, 2, 3, 2, 4, 3, 2, 1 ], [ 1, 2, 3, 2, 1, 4, 3 ] ]

gap> WJp:=Subgroup(W,[W.2,W.3]);
gap> D:=DoubleCosets(W,WI,WJp);
gap> List(D,j->List(Elements(j),i->CoxeterWord(W,i)));
[ [ [  ], [ 4 ], [ 4, 3 ], [ 3 ], [ 2, 1 ], [ 2, 1, 4 ], [ 2, 1, 3, 2 ],
[ 2, 1, 4, 3, 2 ], [ 4, 3, 2 ], [ 3, 2 ], [ 2 ], [ 2, 4 ], [ 2, 1, 3 ],
[ 2, 1, 4, 3 ], [ 2, 4, 3, 2 ], [ 2, 3, 2 ], [ 2, 3 ], [ 2, 4, 3 ],
[ 1 ], [ 1, 4 ], [ 1, 4, 3 ], [ 1, 3 ], [ 1, 2, 1 ], [ 1, 2, 1, 4 ],
[ 1, 2, 1, 3, 2 ], [ 1, 2, 1, 4, 3, 2 ], [ 1, 2 ], [ 1, 2, 4 ],
[ 1, 4, 3, 2 ], [ 1, 3, 2 ], [ 1, 2, 1, 3 ], [ 1, 2, 1, 4, 3 ],
[ 1, 2, 3 ], [ 1, 2, 4, 3 ], [ 1, 2, 4, 3, 2 ], [ 1, 2, 3, 2 ] ],
[ [ 3, 4, 3 ], [ 3, 4 ], [ 2, 1, 3, 2, 4, 3 ], [ 2, 1, 3, 2, 4, 3, 2 ],
[ 3, 4, 3, 2 ], [ 3, 2, 4 ], [ 2, 1, 3, 2, 4 ], [ 2, 1, 3, 4, 3, 2 ],
[ 3, 2, 4, 3, 2 ], [ 3, 2, 4, 3 ], [ 2, 1, 3, 4 ], [ 2, 1, 3, 4, 3 ],
[ 2, 3, 2, 4, 3, 2 ], [ 2, 3, 2, 4, 3 ], [ 2, 3, 4, 3, 2 ],
[ 2, 3, 2, 4 ], [ 2, 3, 4 ], [ 2, 3, 4, 3 ], [ 1, 3, 4, 3 ],
[ 1, 3, 4 ], [ 1, 2, 1, 3, 2, 4, 3 ], [ 1, 2, 1, 3, 2, 4, 3, 2 ],
[ 1, 3, 4, 3, 2 ], [ 1, 3, 2, 4 ], [ 1, 2, 1, 3, 2, 4 ],
[ 1, 2, 1, 3, 4, 3, 2 ], [ 1, 3, 2, 4, 3, 2 ], [ 1, 3, 2, 4, 3 ],
[ 1, 2, 1, 3, 4 ], [ 1, 2, 1, 3, 4, 3 ], [ 1, 2, 3, 4 ],
[ 1, 2, 3, 4, 3 ], [ 1, 2, 3, 4, 3, 2 ], [ 1, 2, 3, 2, 4 ],
[ 1, 2, 3, 2, 4, 3, 2 ], [ 1, 2, 3, 2, 4, 3 ] ],
[ [ 4, 3, 2, 1 ], [ 3, 2, 1 ], [ 2, 1, 4, 3, 2, 1 ], [ 2, 1, 3, 2, 1 ],
[ 2, 4, 3, 2, 1 ], [ 2, 3, 2, 1 ], [ 1, 4, 3, 2, 1 ], [ 1, 3, 2, 1 ],
[ 1, 2, 1, 4, 3, 2, 1 ], [ 1, 2, 1, 3, 2, 1 ], [ 1, 2, 4, 3, 2, 1 ],
[ 1, 2, 3, 2, 1 ] ],
[ [ 3, 4, 3, 2, 1 ], [ 3, 2, 1, 4 ], [ 3, 2, 1, 4, 3, 2, 1 ],
[ 3, 2, 1, 4, 3, 2 ], [ 2, 1, 3, 2, 1, 4, 3, 2, 1 ],
[ 2, 1, 3, 2, 1, 4, 3, 2 ], [ 2, 1, 3, 2, 4, 3, 2, 1 ],
[ 2, 1, 3, 2, 1, 4, 3 ], [ 3, 2, 4, 3, 2, 1 ], [ 3, 2, 1, 4, 3 ],
[ 2, 3, 2, 1, 4, 3, 2, 1 ], [ 2, 3, 2, 1, 4, 3, 2 ],
[ 2, 1, 3, 4, 3, 2, 1 ], [ 2, 1, 3, 2, 1, 4 ], [ 2, 3, 2, 4, 3, 2, 1 ],
[ 2, 3, 2, 1, 4, 3 ], [ 2, 3, 4, 3, 2, 1 ], [ 2, 3, 2, 1, 4 ],
[ 1, 3, 4, 3, 2, 1 ], [ 1, 3, 2, 1, 4 ], [ 1, 3, 2, 1, 4, 3, 2, 1 ],
[ 1, 3, 2, 1, 4, 3, 2 ], [ 1, 2, 1, 3, 2, 1, 4, 3, 2, 1 ],
[ 1, 2, 1, 3, 2, 1, 4, 3, 2 ], [ 1, 2, 1, 3, 2, 4, 3, 2, 1 ],
[ 1, 2, 1, 3, 2, 1, 4, 3 ], [ 1, 3, 2, 4, 3, 2, 1 ],
[ 1, 3, 2, 1, 4, 3 ], [ 1, 2, 3, 2, 1, 4, 3, 2, 1 ],
[ 1, 2, 3, 2, 1, 4, 3, 2 ], [ 1, 2, 1, 3, 4, 3, 2, 1 ],
[ 1, 2, 1, 3, 2, 1, 4 ], [ 1, 2, 3, 4, 3, 2, 1 ], [ 1, 2, 3, 2, 1, 4 ],
[ 1, 2, 3, 2, 4, 3, 2, 1 ], [ 1, 2, 3, 2, 1, 4, 3 ] ] ]