Phân tích Bruhat và ứng dụng

T S(T )
T
S(T ) id
T
.
S(T ) T
S(T ) T.
T = {1, . . . , n} S(T ) n
Σ
n
. Σ
n
n
σ ∈ Σ
n
, L(σ)
L(σ) = {(i, j) | 0 < i < j ≤ n σ(i) > σ(j)},
(σ) = |L(σ)|, σ L
+
(σ)
L
+
(σ) = L(σ)  {(i, i) | 0 < i ≤ n}
= {(i, j) | 0 < i ≤ j ≤ n σ(i) ≥ σ(j)}.
σ = (135)(24)
L(σ) = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)}.
(σ) = 7
L(σ) = {{i, j} | (i, j) ∈ L(σ)}.

L(σ) 2 T {1, , n}
σ : T → σT i < j (i, j) ∈ L(σ)
h : L(σ) −→ L(σ)
{i, j} −→ (i, j)
|L(σ)| = (σ)
 =

i<j
(x
i
− x
j
) ∈ Z[x
1
, . . . , x
n
]. Σ
n
Z[x
1
, . . . , x
n
] x
1
, . . . , x
n
. σ ∈ Σ
n
,
σ =


i<j
(x
σ(i)
− x
σ(j)
) = (−1)
(σ)
.
σ −→ (−1)
(σ)
Σ
n
→ {±1}
(σ
−1
) = (σ).
f : L(σ) −→ L(σ
−1
),
(i, j) −→ (σ(j), σ(i)).
(i, j) ∈ L(σ) i < j σ(i) > σ(j) σ
−1
σ(j) = j
σ
−1
σ(i) = i (σ(j), σ(i)) ∈ L(σ
−1
)
g : L(σ

−1
) −→ L(σ),
(k, l) −→ (σ
−1
(l), σ
−1
(k)).
(k, l) ∈ L(σ
−1
) k < l σ
−1
(k) > σ
−1
(l) σσ
−1
(k) = k
σσ
−1
(l) = l (σ
−1
(l), σ
−1
(k)) ∈ L(σ)
f ◦ g = id
L(σ
−1
)
g ◦ f = id
L(σ)
f

|L(σ
−1
)| = |L(σ)| (σ
−1
) = (σ)
i = 1, , n − 1 s
i
(i, i + 1)
σ, τ ∈ Σ
n
L(στ) = L(τ )τ
−1
∗
L(σ),
AB = (A ∪ B)\(A ∩ B) τ
∗
{i, j} = {τ(i), τ (j)}
L(στ) ⊆ L(τ)τ
−1
∗
L(σ)
{i, j} ∈ L(στ) i < j στ(i) > στ(j) τ(i) = k τ(j) = l
i = τ
−1
(k) j = τ
−1
(l)
1 k < l {i, j} ∈ L(τ) k < l σ(k) > σ(l)
{k, l} ∈ L(σ)
τ

−1
∗
{k, l} = {τ
−1
(k), τ
−1
(l)} = {i, j},
{i, j} ∈ τ
−1
∗
L(σ) {i, j} ∈ L(τ)τ
−1
∗
L(σ)
2 k > l {i, j} ∈ L(τ) {i, j} ∈
τ
−1
∗
L(σ) k > l σ(k) > σ(l) {k, l} ∈ L(σ)
τ
−1
∗
{k, l} = {τ
−1
(k), τ
−1
(l)} = {i, j},
{i, j} ∈ τ
−1
∗

L(σ) {i, j} ∈ L(τ)τ
−1
∗
L(σ)
L(στ) ⊆ L(τ)τ
−1
∗
L(σ).
L(τ)τ
−1
∗
L(σ) ⊆ L(στ).
{i, j} ∈ L(τ)τ
−1
∗
L(σ),
1 {i, j} ∈ L(τ) {i, j} ∈ τ
−1
∗
L(σ) {i, j} ∈ L(τ) i < j
τ(i) > τ(j) i = τ
−1
τ(i) j = τ
−1
τ(j) {i, j} ∈ τ
−1
∗
L(σ)
{τ(i), τ(j)} ∈ L(σ) στ(i) > στ (j) {i, j} ∈ L(στ)
2 {i, j} ∈ L(τ) {i, j} ∈ τ

−1
∗
L(σ) {i, j} ∈ L(τ) i < j
τ(i) < τ(j) i = τ
−1
τ(i) j = τ
−1
τ(j) {i, j} ∈ τ
−1
∗
L(σ)
{τ(i), τ(j)} ∈ L(σ) στ(i) > στ(j) {i, j} ∈ L(στ)
L(τ)τ
−1
∗
L(σ) ⊆ L(στ).
σ(k) < σ(k + 1) (σs
k
) = (σ) + 1 σ(k) >
σ(k + 1) (σs
k
) = (σ) − 1
L(s
k
) = {{k, k + 1}}
(σs
k
) =






(σ) − 1 {k, k + 1} ∈ s
−1
k∗
L(σ)
(σ) + 1 .
s
k∗
{k, k + 1} = {k, k + 1} {k, k + 1} ∈ s
−1
k∗
L(σ) {k, k + 1} ∈
L(σ) σ(k) > σ(k + 1)
(σ) σ (σ)
σ = s
i
1
. . . s
i
r
(σ) ≤ r
(σ) = r r
r = 0 σ
r
r
r > 0 σ k σ(k) > σ(k + 1)
(σs
k

) = r − 1 σs
k
= s
i
1
. . . s
i
r
−1
s
i
1
, . . . , s
i
r
−1
σ = s
i
1
. . . s
i
r
−1
s
k
w = s
i
1
. . . s
i

r
s
1
, . . . , s
n−1
(π(w)) = r w
ρ ρ(i) = n + 1 − i ρ
2
= 1 ρ(i) > ρ(j)
i < j (ρ) =
n(n−1)
2
1 ≤ m < k ≤ n − 1 t
k
m
= s
m
s
m+1
. . . s
k−1
t
k
m
= (m, m + 1, . . . , k − 1, k) t
m
m
= id.
σ ∈ Σ
n

m
1
, m
2
, . . . , m
n
1 ≤ m
k
≤ n
σ = t
n
m
n
t
n−1
m
n−1
. . . t
2
m
2
t
1
m
1
.
(σ) =
n

k=1

(k − m
k
).
m
n
= σ(n) τ = (t
n
m
n
)
−1
σ τ(n) = n
τ Σ
n−1
n n = 1 n − 1
τ = t
n−1
m
n−1
. . . t
2
m
2
t
1
m
1
m
1
, m

2
, . . . , m
n−1
(τ) =
n−1

k=1
(k − m
k
)
σ = t
n
m
n
τ = t
n
m
n
t
n−1
m
n−1
. . . t
2
m
2
t
1
m
1

.
L(t
n
m
n
) = {{i, n} | m
n
≤ i < n}
τ
−1
∗
L(t
n
m
n
) = {{τ
−1
(i), n} | m
n
≤ i < n}.
L(τ) {τ
−1
(i), n} L(τ) τ
−1
∗
L(t
n
m
n
)

(σ) = |L(τ)| + |τ
−1
∗
L(t
n
m
n
)|
= (τ) + (n − m
n
)
=
n

k=1
(k − m
k
).
σ(n) m
1
, m
2
, . . . , m
n
σ = (135)(24) σ = t
5
1
t
4
1

t
3
3
t
2
2
t
1
1
t
5
1
t
4
1
t
3
3
t
2
2
t
1
1
= (12345)(1234)
= (135)(24)
= σ.
˜
Σ
n

s
1
, . . . , s
n
s
i
2
= 1,
s
i
s
i+1
s
i
= s
i+1
s
i
s
i+1
s
i
s
j
= s
j
s
i
|i − j| > 1.
Σ

n
ε :
˜
Σ
n
−→ Σ
n
s
i
s
i
.
t
k
m
= s
m
s
m+1
s
k
˜
Σ
n
Σ
n
ε :
˜
Σ
n

−→ Σ
n
X
n
⊆
˜
Σ
n
t
n
m
n
t
n−1
m
n−1
. . . t
2
m
2
t
1
m
1
1 ≤ m
k
≤ k |X
n
| ≤ n! ε|
X

n
:
˜
Σ
n
−→ Σ
n
X
n
=
˜
Σ
n
X
n
=
n

m=1
t
n
m
X
n−1
X
n−1
=
˜
Σ
n−1

X
n
˜
Σ
n−1
˜
Σ
n
˜
Σ
n−1
s
n−1
X
n
s
n−1
˜
Σ
n−1
t
k
m
k < n − 1
k ≤ n l < n
˜
Σ
n
t
n

k
t
n−1
l
s
n−1
= t
n
k
t
n
l
=





t
n
l+1
t
n−1
l
k ≤ l
t
n
l
t
n−1

k−1
k > l.
t
n
k
t
n−1
l
s
n−1
= t
n
k
t
n
l
l < n
(t
n
l+1
)
−1
t
n
l
= t
n−1
l
(t
n

l
)
−1
.
l = 5 n = 10 k s
k
e
(t
10
6
)
−1
t
10
5
= 987656789 ( 656 = 565)
= 987565789 ( [5, 7] = [5, 8] = [5, 9] = e)
= 598767895 ( 767 = 676)
= 598676895 ( [6, 8] = [6, 9] = e)
= 5698787965 ( 878 = 787)
= 567989765 ( [7, 9] = e)
= 567898765 ( 989 = 898)
= t
9
5
(t
10
5
)
−1

.
l = n − 1
l
s
l+1
s
l
s
l+1
= s
l
s
l+1
s
l
.
t
n
l+2
(t
n
l+1
)
−1
s
l+1
t
n
l+2
= t

n
l+1
(t
n
l+2
)
−1
s
l+1
= (t
n
l+1
)
−1
s
l
s
l+1
t
n
l+2
= t
n
l
.
s
l
t
n
l+2

(t
n
l+2
)
−1
s
l+1
s
l
s
l+1
t
n
l+2
= (t
n
l+1
)
−1
t
n
l
.
(t
n
l+1
)
−1
t
n

l
= s
l
(t
n
l+2
)
−1
s
l+1
t
n
l+2
s
l
.
s
l+1
t
n
l+2
= t
n
l+1
s
l
(t
n
l+2
)

−1
s
l+1
t
n
l+2
s
l
= s
l
t
n−1
l+1
(t
n
l+1
)
−1
s
l
(t
n
l+1
)
−1
t
n
l
= t
n−1

l
(t
n
l
)
−1
.
k ≤ l t
m
k
= t
k
l
t
m
l
t
k
l
t
m
l+1
(t
n
l+1
)
−1
t
n
l

= t
n−1
l
(t
n
l
)
−1
t
l
k
(t
n
k+1
)
−1
t
n
k
= t
n−1
k
(t
n
l
)
−1
.
k ≤ l
t

n
k
t
n
l
= t
n
l+1
t
n−1
l
.
n ≥ i > j k = j l = i − 1
(t
n
l+1
)
−1
t
n
l
= t
n−1
l
(t
n
l
)
−1
(t

n
i
)
−1
t
n
j
= t
n−1
j
(t
n
i−1
)
−1
.
t
n−1
j
(t
n
i−1
)
−1
= t
n−1
j
(t
n−1
i−1

s
n−1
)
−1
= t
n−1
j
s
n−1
(t
n−1
i−1
)
−1
= t
n
j
(t
n−1
i−1
)
−1
,
(t
n
i
)
−1
t
n

j
= t
n
j
(t
n−1
i−1
)
−1
.
t
n
i
t
n
j
= t
n
j
t
n−1
i−1
.
W
r
r s
1
, . . . , s
n
W = 

r
W
r
s
1
, . . . , s
n
∼
r
W
r
us
i
s
j
v ∼ us
j
s
i
v |i − j| > 1,
us
i
s
j
s
i
v = us
j
s
i

s
j
v |i − j| = 1
M
r
= W
r
/ ∼
r
M = 
r
M
r
M
W
s
i
2
= 1,
s
i
s
i+1
s
i
= s
i+1
s
i
s

i+1
,
s
i
s
j
= s
j
s
i
|i − j| > 1.
Σ
n
= M/ < s
2
i
= 1 | i = 1, . . . , n − 1 >
W M
Σ
n
✲
π
❅
❅
❅❘
π



✠

π

.
w = s
p
1
. . . s
p
r
∈ W (π(w)) = r
R
r
r R = 
r
R
r
u ∈ R
r
v ∈ W
r
u ∼
r
v v ∈ R
r
σ, τ ∈ Σ
n
σ τ
ρ ∈ Σ
n
σ = ρτ (σ) = (ρ) + (τ ).

(στ
−1
) = (σ) − (τ ).
u, v π(u) = τ π(uv) = σ.
L(τ) ⊆ L(σ).
σ ∈ Σ
n
σ s
i
s
j
i = j
|i − j| > 1 σ s
i
s
j
= s
j
s
i
.
|i − j| = 1 σ s
i
s
j
s
i
= s
j
s

i
s
j
.
σ s
i
L(s
i
) = {{i, i + 1}} ⊂ L(σ) σ(i) >
σ(i + 1). σ(j) > σ(j + 1).
|i − j| > 1
L(s
i
s
j
) = {{i, i + 1}, {j, j + 1}},
L(s
i
s
j
) ⊆ L(σ).
σ s
i
s
j
.
|i − j| = 1 j = i + 1 σ(i) > σ(i + 1) = σ(j) > σ(j + 1)
{{i, i + 1}, {i + 1, i + 2}, {i, i + 2}} ⊆ L(σ).
τ = s
i

s
j
s
i
= s
j
s
i
s
j
(i, i + 1, i + 3) 3
L(τ) = {{i, i + 1}, {i + 1, i + 2}, {i, i + 2}}.
σ τ
u, v ∈ R
r
π(u) = π(v) u ∼
r
v π

(u) =
π

(v)
r
r = 1 u = s
i
i v = s
j
j
u ∼

r
v
r > 1 u = xs
i
, v = ys
j
x, y ∈ R
r−1
i, j ∈
{1, . . . , n − 1} σ = π(u) = π(v) σ = π(x)s
i
= π(y)s
j
i = j
π(x) = π(y) x, y ∈ R
r−1
x ∼ y,
u = xs
i
∼
r
ys
i
= ys
j
= v.
|i − j| = 1 σ s
i
σ s
i

s
j
s
i
= s
j
s
i
s
j
, z ∈ R
r−3
π(zs
i
s
j
s
i
) = π(zs
j
s
i
s
j
).
|i − j| = 1
u = xs
i
∼ zs
i

s
j
s
i
v = ys
j
∼ zs
j
s
i
s
j
.
zs
i
s
j
s
i
∼ zs
j
s
i
s
j
u = v
|i − j| > 1 σ s
i
σ s
i

s
j
= s
j
s
i
z ∈ R
r−2
π(zs
i
s
j
) = π(zs
j
s
i
).
|i − j| = 1
u = xs
i
∼ zs
j
s
i
v = ys
j
∼ zs
i
s
j

.
zs
i
s
j
∼ zs
j
s
i
u = v
F
n
p
(e
1
, . . . , e
n
) F
n
p
E
i
= F
p
{e
1
, . . . , e
i
} G = GL
n

(F
p
)
T = {g ∈ G | ge
i
∈ F
p
{e
i
} ∀ i}
= }
= {g ∈ G | g
ij
= 0, ∀ i = j},
B = {g ∈ G | gE
i
= E
i
∀ i}
= }
= {g ∈ G | g
ij
= 0, ∀ i > j},
U = {g ∈| ge
i
= e
i
(mod E
i−1
) ∀ i}

= }
= {g ∈ B | g
ii
= 1, ∀ i}.
x ∈ F
p
{e
1
, . . . , e
n
} (gx)
i
=
n

j=1
g
ij
x
j
σ ∈ Σ
n
σ(i) = j e
j
i
σ = (123)
σ =






0 0 1
1 0 0
0 1 0





.
Σ
n
G σe
i
= e
σ(i)
(σx)
i
= x
σ
−1
(i)
ij σ σ
ij
= δ
i,σ(j)
|T | = (p − 1)
n
,

|U| = p
n(n−1)
2
,
|B| = (p − 1)
n
p
n(n−1)
2
,
|G| = p
n(n−1)
2
(p
n
− 1) . . . (p − 1).
n × n T F
p
t = (t
ij
) ∈ T t
ii
= 0 t
ij
= 0 i = j
t = (t
ij
) F
p
, F

p
p − 1
|T | = (p − 1)
n
B
(p − 1)p
n−1
(p − 1)p
n−2
· · ·
n (p − 1)
|B| = (p − 1)p
n−1
(p − 1)p
n−2
(p − 1) . . . (p − 1)
= (p − 1)p
n−1+n−2+···+1
= (p − 1)
n
p
n(n−1)
2
.
|U| = p
n(n−1)
2
|G| n × n GL
n
(F

p
)
F
p
F
p
F
n
p
.
p
n
− 1 1 < k ≤ n k
F
n
p
p
k
− 1 k − 1
p
k
− p
k−1
k
|G| = p
n(n−1)
2
(p
n
− 1) . . . (p − 1).

1 B = U  T.
U ∩ T = {e} B = U  T
U B B = UT.
ϕ : B −→ T b ∈ B t ∈ T





t
ii
= b
ii
t
ij
= 0 i = j.
ϕ kerϕ = {b ∈ B | ϕ(b) = e} = U.
U B ϕ|
T
ϕ(bϕ(b)
−1
) = ϕ(b)ϕ(b)
−1
= e bϕ(b)
−1
∈ kerϕ = U b ∈ Uϕ(b) ⊆ UT
|B| = |UT | B = U  T
n > 1 T B B
U T
C Ob(C)

A, B Ob(C) Mor
C
(A, B) A B
f : A −→ B (f, g) ∈ Mor
C
(A, B) × Mor
C
(B, C),
g ◦ f ∈ Mor
C
(A, C). g ◦ f
f g
(F
1
) f ∈ Mor
C
(A, B) g ∈ Mor
C
(B, C)
h ∈ Mor
C
(C, D)
(h ◦ g) ◦ f = h ◦ (g ◦ f)
A, B, C, D Ob(C).
(F
2
) A Ob(C) id
A
∈ Mor
C

(A, A)
B ∈ Ob(C)
id
B
◦ f = f = f ◦ id
A
, ∀f ∈ Mor
C
(A, B).
(F
3
) Mor
C
(A, B) Mor
C
(C, D)
A = C B = D
(a)
F
(b) Z
(p)
A.
(c) G G
G
(d) n n F
p
V.
(a) Φ C D
Φ : C −→ D A ∈ Ob(C) Φ(A) ∈ Ob(D)
f ∈ Mor

C
(A, B) Φ(f) ∈ Mor
D
(Φ(A), Φ(B))
HT 1. A C Φ(id
A
) = id
Φ(A)
.
HT 2. f : A −→ B g : B −→ C C
Φ(g ◦ f) = Φ(g) ◦ Φ(f).
(b) Φ C D Φ :
C −→ D, A ∈ Ob(C) Φ(A) ∈ Ob(D)
f ∈ Mor
C
(A, B) Φ(f) ∈ Mor
D
(Φ(B), Φ(A))
P B1. A C Φ(id
A
) = id
Φ(A)
.
P B2. f : A −→ B g : B −→ C C
Φ(g ◦ f) = Φ(f ) ◦ Φ(g).
V F n
U = (0 = U
0
< U
1

· · · < U
n
= V)
V.
F lag(V) = { V}.
F lag : V −→ F
V −→ F lag(V),
Φ Ψ C −→ D.
H Φ Ψ X
Ob(C) H
X
Mor
D
(Φ(X), Ψ(X))
Φ(X)
H
X
−−−→ Ψ(X)
Φ(f)






Ψ(f)
Φ(Y ) −−−→
H
Y
Ψ(Y )

f Mor
C
(X, Y ) H
Y
◦ Φ(f) = Ψ(f) ◦ H
X
.
Φ Ψ Nat(Φ, Ψ).
C F
A Ob(C), h
A
(X) = Mor
C
(A, X) X
Ob(C). h
A
C −→ F. F : C −→ F
Nat(h
A
, F )
∼
=
F (A).
H : h
A
−→ F
f : A −→ X,
h
A
(A)

H(A)
−−−→ F (A)
h
A
(f)






F (f )
h
A
(X) −−−→
H(X)
F (X)
.
h
A
(A) = Mor
C
(A, A) id
A
∈ Mor
C
(A, A).
φ : Nat(h
A
, F ) −→ F (A)

H −→ H(A)(id
A
).
u ∈ F (A) g ∈ Mor
C
(A, X),
H : h
A
−→ F
g −→ H(X)(g) = F(g)(u),
H(A) : h
A
(A) = Mor
C
(A, A) −→ F (A)
id
A
−→ H(A)(id
A
) = u,
H
h
A
(A)
H(A)
−−−→ F (A)
h
A
(g)







F (g)
h
A
(X) −−−→
H(X)
F (X)
h
A
(f)






F (f )
h
A
(Y ) −−−→
H(Y )
F (Y )
Y ∈ Ob(C) f ∈ Mor
C
(X, Y ). H ∈ Nat(h
A

, F ). φ
V F
Base(V) = { V} = Mor
V
(F
n
p
, V).
Base : V −→ Flag
V −→ Base(V).
Base
Nat(Base)
∼
=
Base(F
n
p
) = Mor
V
(F
n
p
, F
n
p
) = G.
n F
p
W F lag(W)
W,

U = (0 = U
0
< U
1
< · · · < U
n
= W),
U
i
W, dimU
i
= i. (e
1
. . . e
n
)
F
n
p
, F lag(F
n
p
)
E = (0 = E
0
< E
1
< · · · < E
n
= F

n
p
),
E
i
= F
p
{e
1
, . . . e
i
} E
U, V F lag(W) 0 < i, j ≤ n,
Q
ij
Q
ij
= (U
i
∩ V
j
)/[(U
i−1
∩ V
j
) + (U
i
∩ V
j−1
)].

Q
ij
∼
=
(U
i−1
+ U
i
∩ V
j
)/(U
i−1
+ (U
i
∩ V
j−1
))
f : U
i
∩V
j
−→ (U
i−1
+ U
i
∩ V
j
)/[U
i−1
+ (U

i
∩ V
j−1
)].
Ker f = {x | x ∈ (U
i
∩ V
j
) x ∈ U
i−1
+ (U
i
∩ V
j−1
)}
(U
i−1
∩ V
j
) + (U
i
∩ V
j−1
) ⊆ Ker f.
x ∈ U
i−1
+ (U
i
∩ V
j−1

) x = a + b a ∈ U
i−1
b ∈ (U
i
∩ V
j−1
) x ∈ (U
i−1
∩ V
j
) + (U
i
∩ V
j−1
)
Ker f ⊆ (U
i−1
∩ V
j
) + (U
i
∩ V
j−1
).
Ker f = (U
i−1
∩ V
j
) + (U
i

∩ V
j−1
).
Q
ij
∼
=
(U
i−1
+ U
i
∩ V
j
)/[U
i−1
+ (U
i
∩ V
j−1
)].
(U
i−1
+ U
i
∩ V
0
)/U
i−1
⊂ (U
i−1

+ U
i
∩ V
1
)/U
i−1
⊂ · · · ⊂ (U
i−1
+ U
i
∩ V
n
)/U
i−1
,
U
i
/U
i−1
n

j=1
(U
i−1
+ U
i
∩ V
j
)/U
i−1

= U
i
/U
i−1
.
Q
i,1
, . . . , Q
i,n
U
i
/U
i−1
. i, j = σ(i) Q
i,σ(i)
= 0
j, i = τ (j) Q
τ(j),j
= 0.
σ τ Σ
n
σ
U V δ(U, V ) = σ δ(V , U) = δ(U, V )
−1
δ(V , U),
σ ∈ Σ
n
, δ(σE, E) = σ E
U
i

= σE
i
= F
p
{e
σ(1)
, . . . , e
σ(i)
} = F
p
{e
k
| σ
−1
(k) ≤ i},
V
j
= E
j
= F
p
{e
1
, . . . , e
j
}.
A
i
= {e
k

| σ
−1
(k) ≤ i} B
j
= {e
k
| k ≤ j}. A
i
∩ B
j
U
i
∩ V
j
,
C
ij
= (A
i
∩ B
j
)\[(A
i−1
∩ B
j
) ∪ (A
i
∩ B
j−1
)],

Q
ij
C
ij
= (A
i
\A
i−1
) ∩ (B
j
\B
j−1
) = {e
σ(i)
∩ e
j
}
Q
ij
= 0 j = σ(i) δ(σE, E) = σ
W = F
5
p
a, b, c, d, e, f, g ∈ F
p
u
1
=












a
b
c
d
1











; u
2
=












e
f
g
1
0











; u
3
=












1
0
0
0
0











; u
4
=












0
1
0
0
0











; u
5
=












0
0
1
0
0











.
U = (0 = U
0
< U
1

< U
2
< U
3
< U
4
< U
5
= F
5
p
), U
i
=
F
p
{u
1
, . . . , u
i
}, i = 1, . . . , 5. Q
15
u
1
Q
24
u
2
Q
31

u
3
Q
42
u
4
Q
53
u
5
Q
ij
= 0
{i, j} δ(U, E) = (1, 5, 3)(2, 4)
δ(gU, gV ) = δ(U, V ) U, V ∈ F lag(W) g ∈ Aut(W)
δ(U, V ) = σ, Q
ij
= 0 j = σ(i)
gQ
ij
= (gU
i
∩ gV
j
)/(gU
i−1
∩ gV
j
) + (gU
i

∩ gV
j−1
),
gQij = 0 Q
ij
= 0 j = σ(i) δ(gU, gV ) = σ
δ(gU, gV ) = δ(U, V )
T , U, V Flag(W)
δ(T, V ) = δ(T , U)δ(U, V ).
n = 2 F lag(W)
W
ρ ∈ Σ
2
, ρ = e δ(L, L) = e δ(L, M) = ρ L = M
L, M N Flag(W) δ(L, N) = δ(L, M)δ(M, N)
δ(U, V ) s
i
U, V ∈ F lag(W) δ(U, V ) = σ, 0 < i  n. U
i−1
=
V
i−1
σ(i) = i U
i
= V
i
A = U
i−1
= V
i−1

(i − 1)
U
i−1
∩ V
i
= V
i−1
∩ V
i
= A,
U
i
∩ V
i−1
= A Q
ii
= U
i
∩ V
i
/A σ(i) = i
Q
ii
= 0 Q
ii
= 0 U
i
∩ V
i
⊃ A U

i
V
i
i
A i − 1 U
i
∩ V
i
⊃ A U
i
= V
i
δ(U, V ) = e U = V
U = V δ(U, V ) = e.
U
0
= V
0
U
1
= V
1
U
i
= V
i
i = 1, 2, . . . , n U = V .
δ(U, V ) = s
i
U

j
= V
j
j = i U
i
= V
i
δ(U, V ) = s
i
. U
j
= V
j
j < i,
U
i
= V
i
.
A = U
i−1
= V
i−1
. U
i
= V
i
U
i
∩V

i
⊂ U
i
U
i
∩V
i
⊃ A.
U
i
∩ V
i
= A Q
i,i+1
= (U
i
∩ V
i+1
)/A δ(U, V ) = s
i
Q
i,i+1
= 0
dim(U
i
∩ V
i+1
) ≥ dim(A) + 1 = i.
dim(U
i

) = i U
i
∩ V
i+1
= U
i
U
i
⊂ V
i+1
V
i
⊂ V
i+1
dim(U
i
+ V
i
) = dim(U
i
) + dim(V
i
) − dim(U
i
∩ V
i
),
= i + 1 = dim(V
i+1
),

V
i+1
= U
i
+ V
i
U
i+1
=
U
i
+ V
i
j > i + 1 s
i
(j) = j U
j
= V
j
j > i
V ∈ F lag(W) σ ∈ Σ
n
, n
Y = Y (σ, V ) = {U | δ(U, V ) = σ} ⊂ Flag(W).
W = F
n
p
Y (σ) = Y (σ, E) ⊂ Flag(F
n
p

).
X = X(σ) = U ∩ U
(σρ)
−1
 U,
ρ(i) ρ(i) = n − i + 1, U
σ
= σ
−1
Uσ.
g = (g
ij
) ∈ X(σ)
g
ij
=













1 i = j,

(i, j) ∈ L(σ
−1
),
0 .
|X(σ)| = p
(σ)
g ∈ U g
ii
= 1 g
ij
= 0
i > j
g ∈ U
τ
τgτ
−1
∈ U
(τgτ
−1
)
ij
=

k,l≥1
τ
ik
g
kl
τ
−1

lj
.
τ
ik
= δ
i,τ(k)
τ
−1
ik
= δ
i,τ
−1
(k)
,
(g
τ
−1
)
ij
= g
τ
−1
(i)τ
−1
(j)
= g
kl
,
k = τ
−1

(i), l = τ
−1
(j) τgτ
−1
∈ U (τgτ
−1
)
ij
= 0 i > j,
g
kl
= 0 τ(k) > τ(l)
g ∈ U
(σρ)
−1
= U
ρσ
−1
g
ij
= 0 i, j
ρσ
−1
(i) > ρσ
−1
(j) i, j σ
−1
(i) < σ
−1
(j) ρ

g
ij
= 0 σ
−1
(i) > σ
−1
(j) i < j
g
ij
(i, j) ∈ L(σ
−1
)
X(σ) = U ∩ U
(σρ)
−1
 U,
φ : X(σ) −→ Y (σ)
g −→ gσE.
|Y (σ)| = |Y (σ, V )| = p
(σ)
V ∈ F lag(W)
g ∈ X(σ) gσE ∈ Y (σ)
X(σ) ⊆ U ⊆ B g
−1
E = E
δ(gσE, E) = δ(σE, g
−1
E) = δ(σE, E) = σ.
φ σE B
σ

−1
g ∈ G σE gσE = σE
σ
−1
gσE = E. σ
−1
gσ ∈ B g ∈ B
σ
−1
σE
X(σ) = U ∩ U
(σρ)
−1
H = B
σ
−1
∩ U
(σρ)
−1
= (B ∩ U
ρ
)
σ
−1
.
B ∩U
ρ
= {e} B U
ρ
X(σ)

σE,
φ : X(σ) −→ Y (σ),
φ
T
i
= F
p
{e
m
| m ≤ σ(i), σ
−1
(m) ≥ i} ⊆ E
σ(i)
,

i
: F
n
p
−→ F
p
i V ∈ Y (σ)
v
i
∈ V
i
∩ T
i

σ(i)

(v
i
) = 1
(v
1
, . . . , v
i
) V
i
F
p
i i = 1
T
1
= F
p
{e
m
| m ≤ σ(1), σ
−1
(m) ≥ 1} ⊆ E
σ(1)
.
V ∈ Y (σ) Q
1j
= 0 j = σ(1)
Q
1j
= (V
1

∩ E
j
)/((V
1−1
∩ E
j
) + (V
i
∩ E
j−1
))
= V
1
∩ E
j
/(V
1
∩ E
j−1
)
= V
1
∩ E
σ(1)
/(V
1
∩ E
σ(1)−1
).
Q

1j
= 0 V
1
∩ E
σ(1)−1
= ∅ V
1
1
v
1
V
1
v
1
∈ E
σ(1)
V
1
∩ E
σ(1)−1
= ∅
v
1
∈ E
σ(1)−1
v
1
= e
σ(1)
v

1
∈ T
1
∩ V
1

σ(1)
(v
1
) = 1
i = 1
j < i
S
i
= F
p
{v
j
| j ≤ i, σ(j) < σ(i)} ⊆ V
i
∩ E
σ(i)
.