Antipodal sets of Grassmann manifolds and their quotient spaces

グラスマン多様体とその商空間の極大対蹠 集合

Antipodal sets of Grassmann

manifolds and their quotient spaces

田中 真紀子（東京理科大学）

第

2

2019

6

22

–6

23

茨城大学

Joint work with Hiroyuki Tasaki.

1. Antipodal sets and 2-numbers

2. Maximal antipodal sets of Grassmann manifolds (1)

3. Maximal antipodal sets of Grassmann manifolds (2)

4. The quotient space G _m ( K ^{2 m} ) ^∗ ( K = R , C , H ) 5. Maximal antipodal sets of G _m ( K ^{2 m} ) ^∗

6. The 2-number and great antipodal sets

of G _m ( K ^{2 m} ) ^∗

1. Antipodal sets and 2-numbers

M : compact Riemann symmetric space s _x : geodesic symmetry at x

i.e., (i) s _x is an isometry of M , (ii) s _x ² = id, (iii) x is an isolated fixed point of s _x

S ⊂ M : subset

S : antipodal set ⇐⇒ ∀ ^def x, y ∈ S, s _x ( y ) = y 2-number of M

# ₂ M := max {| S | | S ⊂ M antipodal set }

S : great antipodal set ⇐⇒ | ^def S | = # ₂ M

(Chen-Nagano 1988)

Examples. (1) M = S ⁿ ( ⊂ R ^{n +1} )

{ x, − x } ( x ∈ S ⁿ ) : great antipodal set

# ₂ S ⁿ = 2

(2) M = R P ⁿ

e ₁ , . . . , e _{n +1}

o.n.b. of R ^{n +1}

{⟨ e ₁ ⟩ _R , . . . , ⟨ e _{n +1} ⟩ _R } : great antipodal set

# ₂ R P ⁿ = n + 1

(3) M = U ( n ) s _x ( y ) = xy ^{− 1} x

s ₁

( y ) = y ⇔ y ² = 1 _n (1 _n : identity matrix)

x ² = y ² = 1 _n ⇒ s _x ( y ) = y iff xy = yx

∆ _n :=























± 1

. . .

± 1























: great antipodal set

# ₂ U ( n ) = 2 ⁿ

Theorem 1. (T.-Tasaki 2013) M : symmetric R -space

(1) Any antipodal set of M is included in a great antipodal set.

(2) Any two great antipodal sets are I ₀ ( M )-

congruent.

I ₀ ( M ) : identity component of I ( M ), the group of isometries of M

(3) A great antipodal set of M is an orbit of the Weyl group.

S ⁿ , R P ⁿ and U ( n ) are symmetric R -spaces.

A maximal antipodal set is not necessarily

a great antipodal set.

2. Maximal antipodal sets of Grassmann manifolds (1)

O ( n, K ) :=



















O ( n ) ( K = R ) U ( n ) ( K = C ) Sp ( n ) ( K = H )

G _m ( K ⁿ ) : Grassmann manifold of m -dim K - subspaces of K ⁿ

O ( n, K ) y G _m ( K ⁿ ) transitively

G _m ( K ⁿ ) = ∼ O ( n, K ) /O ( m, K ) × O ( n − m, K )

G _m ( K ⁿ ) is a Riemann symmetric space w.r.t.

O ( n, K )-invariant Riemann metric. G _m ( K ⁿ )

is a symmetric R -space.

x ∈ G _m ( K ⁿ )

π _x , π _x

orthogonal projection onto x, x ^⊥ ρ _x := π _x − π _x

: K ⁿ → K ⁿ

s _x ( y ) = ρ _x ( y ) ( y ∈ G _m ( K ⁿ )) e ₁ , . . . , e _n : o.n.b. of K ⁿ

ρ _e

( e _i ) = e _i , ρ _e

( e _j ) = − e _j ( i ̸ = j ) s _{⟨ e}

i1

,...,e

⟩

( ⟨ e _j

, . . . , e _j

⟩ _K ) = ⟨ e _j

, . . . , e _j

⟩ _K

A := {⟨ e _i

, . . . , e _i

⟩ _K | 1 ≤ i ₁ < · · · < i _m ≤ n } is an antipodal set of G _m ( K ⁿ ).

| A | =

(

n m

)

= # ₂ G _m ( K ⁿ ) (Chen-Nagano)

A is a unique great antipodal set up to

I ₀ ( G _m ( K ⁿ ))-congruence by Theorem 1.

3. Maximal antipodal sets of Grassmann manifolds (2)

O ( n, K ) is a Riemann symmetric space w.r.t.

a bi-invariant metric.

x ∈ O ( n, K ), s _x ( y ) = xy ^{− 1} x ( y ∈ O ( n, K ))

1 _n : identity matrix, s ₁

( y ) = y ⇔ y ² = 1 _n F ( s ₁

, O ( n, K )) =

0 ≤ k ≤ n

∪

g ∈ O ( n, K )

g





1 _k

− 1 _{n − k}





g ^{− 1} ι : G _m ( K ⁿ ) ∋ x 7→ ρ _x ∈ O ( n, K ) : embedding

F ( s ₁

, O ( n, K )) =

0 ≤ k ≤ n

ι ( G _k ( K ⁿ ))

G : compact Lie group, e : identity element G ₀ : identity component of G

M : connected component of F ( s _e , G ) M is a polar of G w.r.t. e .

x ∈ M, M = { I _g ( x ) | g ∈ G ₀ } , I _g ( x ) = gxg ^{− 1} I ₀ ( M ) = { I _g | _M | g ∈ G ₀ }

ι ( G _m ( K ⁿ )) =

id G _m ( K ⁿ ) is a polar of O ( n, K ) w.r.t. 1 _n .

A : maximal antipodal set of G _m ( K ⁿ )

{ 1 _n } ∪ A : antipodal set of O ( n, K )

∃ A ˜ : maximal antipodal subgroup of O ( n, K ) s.t. { 1 _n } ∪ A ⊂ A ˜

∃ g ∈ O ( n, K ) s.t. A ˜ = g ∆ _n g ^{− 1}

When K = R , we can take g ∈ SO ( n ).

A = ˜ A ∩ G _m ( K ⁿ )

A = g ∆ _n g ^{− 1} ∩ G _m ( K ⁿ ) = g (∆ _n ∩ G _m ( K ⁿ )) g ^{− 1}

∆ _n ∩ G _m ( K ⁿ )

= ∆ ^m _n :=























ε ₁

. . .

ε _n









∈ ∆ _n

|{ i | ε _i = 1 }| = m















A = g ∆ ^m _n g ^{− 1}

∆ ^m _n is a unique maximal antipodal set (hence a unique great antipodal set) of G _m ( K ⁿ ) up to I ₀ ( G _m ( K ⁿ ))-congruence.

| ∆ ^m _n | =

(

n m

)

= # ₂ G _m ( K ⁿ ) ι ^{− 1} (∆ ^k _n )

= {⟨ e _i

, . . . , e _i

⟩ _K | 1 ≤ i ₁ < · · · < i _m ≤ n }

4. The quotient space G _m ( K ^{2 m} ) ^∗ ( K = R , C , H ) γ : G _m ( K ^{2 m} ) ∋ x 7→ x ^⊥ ∈ G _m ( K ^{2 m} ) : isometry

G _m ( K ^{2 m} ) ^∗ := G _m ( K ^{2 m} ) / { id , γ }

s _{[ x ]} ([ y ]) = [ s _x ( y )] ([ x ] , [ y ] ∈ G _m ( K ^{2 m} ) ^∗ ) O (2 m, K ) ^∗ := O (2 m, K ) / {± 1 _{2 m} }

π _{2 m} : O (2 m, K ) → O (2 m, K ) ^∗ : projection

ι ◦ γ ( x ) = ι ( x ^⊥ ) = ρ _x

= − ρ _x ( x ∈ G _m ( K ^{2 m} )) G _m ( K ^{2 m} ) ^∗ =

id ι ( G _m ( K ⁿ )) / {± 1 _{2 m} } ⊂ O (2 m, K ) ^∗

G _m ( K ^{2 m} ) ^∗ is a polar of O (2 m, K ) ^∗ w.r.t. π _{2 m} (1 _{2 m} ).

5. Maximal antipodal sets of G _m ( K ^{2 m} ) ^∗ D [4] :=











± 1 0 0 ± 1





,





0 ± 1

± 1 0











n = 2 ^k · l, l : odd 0 ≤ s ≤ k

D ( s, n ) := D [4] ⊗ · · · ⊗ D [4]

| {z }

s

⊗ ∆ _{n/ 2}

⊂ O ( n )

= { d ₁ ⊗ · · · ⊗ d _s ⊗ d ₀

| d ₁ , . . . , d _s ∈ D [4] , d ₀ ∈ ∆ _{n/ 2}

}

∆ ₂ ( D [4]

D ( k − 1 , 2 ^k ) = D [4] ⊗ · · · ⊗ D [4]

| {z }

k − 1

⊗ ∆ ₂ ( D [4] ⊗ · · · ⊗ D [4]

| {z }

k − 1

⊗ D [4] = D ( k, 2 ^k )

Theorem 2. (T.-Tasaki 2017)

A maximal antipodal subgroup (MAS) of O ( n, K ) ^∗ is given as follows.

(1) K = R . MAS of O ( n ) ^∗ is O ( n ) ^∗ -conjugate to one of the following.

π _n ( D ( s, n )) (0 ≤ s ≤ k ) ,

where the case ( s, n ) = ( k − 1 , 2 ^k ) is excluded.

(2) K = C . MAS of U ( n ) ^∗ is U ( n ) ^∗ -conjugate to one of the following.

π _n ( { 1 , √

− 1 } D ( s, n )) (0 ≤ s ≤ k ) ,

where the case ( s, n ) = ( k − 1 , 2 ^k ) is excluded.

(3) K = H . MAS of Sp ( n ) ^∗ is Sp ( n ) ^∗ -conjugate to one of the following.

π _n ( { 1 , i , j , k } D ( s, n )) (0 ≤ s ≤ k ) ,

where the case ( s, n ) = ( k − 1 , 2 ^k ) is excluded.

In (1) we can replace “ O ( n ) ^∗ -conjugate” by

“ SO ( n ) ^∗ -conjugate”.

A : maximal antipodal set of G _m ( K ^{2 m} ) ^∗

{ π _{2 m} (1 _{2 m} ) } ∪ A : antipodal set of O (2 m, K ) ^∗

∃ A ˜ : maximal antipodal subgroup of O (2 m, K ) ^∗

s.t. { π _{2 m} (1 _{2 m} ) } ∪ A ⊂ A ˜

A = ˜ A ∩ G _m ( K ^{2 m} ) ^∗ 2 m = 2 ^k · l, l : odd Γ _K :=



















{ 1 } ( K = R ) { 1 , √

− 1 } ( K = C ) { 1 , i , j , k } ( K = H )

∃ g ∈ O (2 m, K ) ( ∃ g ∈ SO (2 m ) when K = R ) ,

∃ s ∈ { 0 , . . . , k } s.t.

A ˜ = π _{2 m} ( g ) π _{2 m} (Γ _K D ( s, 2 m )) π _{2 m} ( g ) ^{− 1}

A = π _{2 m} ( g ) π _{2 m} (Γ _K D ( s, 2 m )) π _{2 m} ( g ) ^{− 1} ∩ G _m ( K ^{2 m} ) ^∗

= π _{2 m} ( g ) π _{2 m} (Γ _K D ( s, 2 m ) ∩ G _m ( K ^{2 m} )) π _{2 m} ( g ) ^{− 1}

P D ( s, 2 m ) := { d ∈ D ( s, 2 m ) | d ² = 1 _{2 m} }

N D ( s, 2 m ) := { d ∈ D ( s, 2 m ) | d ² = − 1 _{2 m} } D ( s, 2 m ) ∩ G _m ( R ^{2 m} )

= { d ∈ D ( s, 2 m ) | d ² = 1 _{2 m} , Tr d = 0 }

= { d ₁ ⊗ · · · ⊗ d _s ⊗ d ₀ ∈ P D ( s, 2 m ) |

∃ d _i (0 ≤ i ≤ s ) Tr d _i = 0 } AG ( s, 2 m ) := π _{2 m} ( D ( s, 2 m ) ∩ G _m ( R ^{2 m} ))

MAS:=maximal antipodal set Theorem 3. (T.-Tasaki)

(1) MAS of G _m ( R ^{2 m} ) ^∗ is SO (2 m ) ^∗ -congruent

to AG ( s, 2 m ) (0 ≤ s ≤ k ) with exceptions ( ∗ ).

(2) MAS of G _m ( C ^{2 m} ) ^∗ is U (2 m ) ^∗ -congruent to AG ( s, 2 m ) ∪ π _{2 m} ( √

− 1 N D ( s, 2 m )) (0 ≤ s ≤ k ) with exceptions ( ∗ ).

(3) MAS of G _m ( H ^{2 m} ) ^∗ is Sp (2 m ) ^∗ -congruent to AG ( s, 2 m ) ∪ π _{2 m} ( { ⁱ , j , k } N D ( s, 2 m )) (0 ≤ s ≤ k ) with exceptions ( ∗ ).

( ∗ ) : AG ( k − 1 , 2 ^k ) when 2 m = 2 ^k and AG (0 , 4) when 2 m = 4.

AG (0 , 4) = π ₄ ( {± I ₁ ⊗ 1 ₂ , ± 1 ₂ ⊗ I ₁ , ± I ₁ ⊗ I ₁ } )

( AG (2 , 4).

6. The 2-number and great antipodal sets of G _m ( K ^{2 m} ) ^∗

AG _C ( s, 2 m ) := AG ( s, 2 m ) ∪ π _{2 m} ( √

− 1 N D ( s, 2 m )) AG _H ( s, 2 m ) := AG ( s, 2 m ) ∪ π _{2 m} ( { ⁱ , j , k } N D ( s, 2 m )) Since N D (0 , 2 m ) = ∅ , AG (0 , 2 m ) = AG _C (0 , 2 m )

= AG _H (0 , 2 m ).

AG ( s, 2 m ) ( AG _C ( s, 2 m ) ( AG _H ( s, 2 m )

(1 ≤ s ≤ k )

| AG (0 , 2 m ) | = | π _{2 m} (∆ ^m _{2 m} ) | =

(

2 m m

)/

2

| AG ( s, 2 m ) |

= 2

m

2s−1

− 1

(2 ^{2 s − 1} + 2 ^{s − 1} − 1) +

(

m/ 2

m/ 2

)/

2

| AG _C ( s, 2 m ) |

= 2

m

2s−1

− 1

(2 ^{2 s} − 1) +

(

m/ 2

m/ 2

)/

2

| AG _H ( s, 2 m ) |

= 2

m

2s−1

− 1

(2 ^{2 s +1} − 2 ^s − 1) +

(

m/ 2

m/ 2

)/

2

(1 ≤ s ≤ k ) We set

(

m/ 2

m/ 2

)

= 0 when m/ 2 ^s ∈ / Z . We can show: when m ≥ 5,

| AG (0 , 2 m ) | > | AG _H ( s, 2 m ) | > | AG _C ( s, 2 m ) | >

| AG ( s, 2 m ) | (1 ≤ s ≤ k ).

We need a case-by-case argument when m ≤ 4.

GAS:=great antipodal set Theorem 4. (T.-Tasaki)

GAS of G _m ( K ^{2 m} ) ^∗ (up to congruence) and

# ₂ G _m ( K ^{2 m} ) ^∗ are as follows.

I. G _m ( R ^{2 m} ) ^∗

(1) m = 1 AG (1 , 2) , # ₂ G ₁ ( R ² ) ^∗ = 2 (2) m = 2 AG (2 , 4) , # ₂ G ₂ ( R ⁴ ) ^∗ = 9

(3) m = 4 AG (0 , 8) , AG (3 , 8) , # ₂ G ₄ ( R ⁸ ) ^∗ =

35

(4) m ̸ = 1 , 2 , 4 AG (0 , 2 m ) , # ₂ G _m ( R ^{2 m} ) ^∗ =

(

2 m m

)

/ 2

II. G _m ( C ^{2 m} ) ^∗

(1) m = 1 AG _C (1 , 2) , # ₂ G ₁ ( C ² ) ^∗ = 3 (2) m = 2 AG _C (2 , 4) , # ₂ G ₂ ( C ⁴ ) ^∗ = 15 (3) m = 3 AG _C (1 , 6) , # ₂ G ₃ ( C ⁶ ) ^∗ = 12 (4) m = 4 AG _C (3 , 8) , # ₂ G ₄ ( C ⁸ ) ^∗ = 63

(5) m ̸ = 1 , 2 , 3 , 4 AG _C (0 , 2 m ) , # ₂ G _m ( C ^{2 m} ) ^∗ =

(

2 m m

)

/ 2

III. G _m ( H ^{2 m} ) ^∗

(1) m = 1 AG _H (1 , 2) , # ₂ G ₁ ( H ² ) ^∗ = 5 (2) m = 2 AG _H (2 , 4) , # ₂ G ₂ ( H ⁴ ) ^∗ = 27 (3) m = 3 AG _H (1 , 6) , # ₂ G ₃ ( H ⁶ ) ^∗ = 20 (4) m = 4 AG _H (3 , 8) , # ₂ G ₄ ( H ⁸ ) ^∗ = 119

(5) m ̸ = 1 , 2 , 3 , 4 AG _H (0 , 2 m ) , # ₂ G _m ( H ^{2 m} ) ^∗ =

(

2 m m

)

/ 2

Thank you for your kind attention.