Flag varieties

Janina Letz October 26, 2015

K a field and V a n-dimensional vector space over K.

1 Grassmannian variety

Definition (Grassmannian).

Gr(r, V ) = {U ⊂ V |r − dimensional subspace}

Proposition. Gr(r, V ) is a projective variety. Proof. Pl¨ucker embedding:

r ^ Φ: Gr(r, V ) → P( (V )) U 7→ K(u1, . . . , ur) where {u1, . . . , ur} a basis of U. 1. Φ well-defined 0 0 0 {u1, . . . , ur} and {u1, . . . , ur} basis of U. Then u1 ∧· · ·∧ur = det(M)u1 ∧ 0 0 · · · ∧ ur with M change of basis matrix. So K(u1 ∧ · · · ∧ ur) = K(u1 ∧ 0 · · · ∧ ur). 2. Φ injective U, W ∈ Gr(r, V ) with basis {u1, . . . , ur} and {w1, . . . , wr} respectively. Let u = u1 ∧ · · · ∧ ur and w = w1 ∧ · · · ∧ wr. Consider

r+1 ^ φu : v → (V ) v 7→ v ∧ w

It is Φ(U) = Φ(W ) ⇔ ∃x ∈ K : u = xw ⇔ ∃x ∈ K : φw = xφu ⇔ ker(φw) = ker(φu). It is enough to show: ker(φw) = W . It is clear, that ker(φw) ⊃ W . For the other direction extend the basis of W to {w1, . . . , wn} a basis of V . P Let v ∈ ker(φw), v = aiwi. Then

n X 0 = v ∧ w = aiwi ∧ w ⇒ ai = 0 i ≥ r + 1 ⇒ v ∈ W i=r+1

1 Vr 3. Φ is closed in P( (V )) (a) v ∈ V, w ∈ Vr(V ). Then v divides w ⇔ v ∧ w = 0 in Vr+1(V ) One direction is clear. For the other choose a basis {v1, . . . , vn} of V with v = v . Then w = P a v ∧ · · · ∧ v . Then for 1 1≤i1<···

0 = v ∧ w it is ai1...ir = 0 if i1 6= 1. So v divides w. Vr (b) w ∈ (V ). Let w1, . . . , wt be a basis of ker(φw). So w = w1 ∧ · · · ∧ Vr−t wt ∧ u for u ∈ . So dim(ker(φw)) ≤ r ⇒ rank(φw) ≥ n − r. On the other hand it is w = w1 ∧ · · · ∧ wr ⇒ rank(φw) = n − r. So: w decomposable ⇔ rank(φw) = n − r ⇔ rank(φw) ≤ n − r. Vr P (c) Let w ∈ (V ) with w = ai1...ir vi1 ∧ · · · ∧ vir . ai1...ir are called Vr dim(Vr (V ))−1 Pl¨ucker coordinates of W . They associate P( (V )) with P . Then

Kw ∈ im(Φ) ⇔ rank(φw) ≤ n − r ⇔ rank(M(φw)) ≤ n − r

for M(φw) = (bjf(k)) n j=1...n,k=1...(r+1) ( 0 j ≤ i ∀k b = k ji1...ir+1 (−1)ka i = j i1...ˆik...ir+1 k n
for f : {1 ... r+1 → {(i1, . . . , ir+1)|1 ≤ i1 < ··· < ir+1} a bijection of sets. So Gr(r, V ) =∼ V ({(n + r − 1)-minor of B = 0}).

It is: Gr(n − r, V ) =∼ Gr(r, V ) and Gr(1,V ) =∼ P(V ) =∼ Pdim(V )−1.

Example. Let V a 4-dimensional vector space with basis {v1, . . . , v4}. • Let W ∈ Gr(1,V ) with basis {w}. X w = aivi

Matrix representation of φw:   a2 −a1 0 0 a3 0 −a1 0    a4 0 0 −a1    0 a3 −a2 0     0 a4 0 −a2 0 0 a4 −a3

3 Gr(1,V ) =∼ V ({0}) =∼ P

2 • Let W ∈ Gr(2,V ) with basis {w1, w2}. X X X w1 = aivi w2 = bivi w1 ∧ w2 = (aibj − ajbi) vi ∧ vj i

Matrix representation of φw1∧w2 :   a23 −a13 a12 0 a24 −a14 0 a12   a34 0 −a14 a13 0 a34 −a24 a23

So there are 16 equations. V2 Another way: Use, that for u ∈ (V ) it is ∃ui ∈ V for u = u1 ∧ u2 ⇔ u ∧ u = 0. So ∼ Gr(2,V ) = V (X12X34 − X13X24 + X14X23)

• Let W ∈ Gr(3,V ) with basis {w1, w2, w3}. X X X w1 = aivi w2 = bivi w3 = civi

X w1∧w2∧w3 = (aibjck + ajbkci + akbicj − akbjci − ajbick − aibkcj) vi∧vj∧vk i

Matrix representation of φw1∧w2∧w3 :
a234 −a134 a124 −a123

3 Gr(3,V ) =∼ V ({}) =∼ P

2 Flag variety

Definition (Flag variety).

F (V ; n1, . . . , nr) = {0 ⊂ V1 ⊂ · · · ⊂ Vr ⊂ V | dim(Vi) = ni} for 1 ≤ n1 < ··· < nr ≤ n a sequence of integers.

Proposition. F (V ; n1, . . . , nr) is a projective variety. Proof.

Ψ: F (V ; n1, . . . , nr) → Gr(n1,V ) × · · · × Gr(nr,V ) {Vi} 7→ (V1,...,Vr) is an embedding. Show the image is closed.

3 1. Consider the projection πij : Gr(n1,V ) × · · · × Gr(nr,V ) → Gr(ni,V ) × Gr(nj,V ) for i < j. Then

\ −1 Ψ(F (V ; n1, . . . , nr)) = πij (πij(Ψ(F (V ; n1, . . . , nr)))) i

So it is enough to prove Ψ(F (V ; r, s)) is closed in Gr(r, V ) × Gr(s, V ) for r < s.

2. (U, W ) ∈ Gr(r, V ) × Gr(s, V ) and {u1, . . . , ur) and {w1, . . . , ws} basis of U and W and u = u1 ∧ · · · ∧ ur and w = w1 ∧ · · · ∧ ws. Then

r+1 s+1 ^ ^ φu ⊕ φw : V → (V ) ⊕ (V ) ker(φu ⊕ φw) = U ∩ W

So rank(φu ⊕ φw) = n − dim(U ∩ W ) ≥ n − r. This means

U ⊂ W ⇔ rank(φu ⊕ φw) = n − r ⇔ rank(φu ⊕ φw) ≤ n − r

This gives polynomial equations in the elements of the matrices of φu and φw. So F (v; r, s) is a projective variety.

Example. Let V be a 4-dimensional vector space. (U, W ) ∈ F (V ; 1, 2), {u} P P and {w1, w2} basis of U and W . u = aivi and w1 ∧ w2 = i

3 Algebraic groups

Definition (Algebraic group). let G be an algebraic variety with two morphism of algebraic varieties m : G × G → G and i : G → G, such that G is a group with multiplication m and i as inverse map. Then G is an algebraic group.

n2 ∼ n2+1 Example. Gl(n, k) = A \V (det) = V (Xn2−1 det −1) ⊂ A via (a1, . . . , an2 ) 7→ 1 (a1, . . . , a 2 , ). Then the multiplication is matrix multiplication in the n det((ai)) first n2 coordinates and multiplication of the last coordinate in the last coor- dinate. The inverse map is the last component times the adjugate of the first n2 components and the determinant of the first n2 coordiantes in the last one. Since these maps are polynomials, they are morphisms of algebraic varieties.

4 Definition (Linear). An algebraic group G is linear, if G ≤ Gl(n, k). Lemma. Let G be an algebraic group. Then G is linear ⇔ G is affine.

Example. The upper triangle matrices: T (n, k) = V ({Xij|i < j}) ∩ Gl(n, k) ⊂ Gl(n, k). From now G is a linear algebraic group. Definition. G an algebraic group and X an algebraic variety. Then G acts on X, if there is a map G × X → X, that is a group action and a morphism of algebraic varieties. Example. • Gl(V ) acts on Gr(r, V ): (A, W ) 7→ A(W )

• Gl(V ) acts on Gr(n1,V ) × · · · × Gr(nr,V )

• Gl(V ) acts on F (V ; n1, . . . , nr) Proposition. Gl(V ) acts transitively on flag varieties.

Proof. Let {Uni }, {Wni } ∈ F (V ; n1, . . . , nr). Then choose basis {u1, . . . , un} and {w1, . . . , wn} of V , such that {u1, . . . , uni } a basis of Uni and {w1, . . . , wni } basis of Wni . Then define A ∈ Gl(V ) as A(ui) = wi.

Let {Vi} ∈ F (V ; 1, . . . , n) and {v1, . . . , vn} a basis corresponding to {Vi}. ∼ Then Gl(V ) = Gl(n, K) by the basis and the stabilizer of {Vi} is isomorphic to T (n, K). If H is a closed subgroup of G, then G/H is an algebraic variety with a canonical projection π : G → G/H, st for a morphism of algebraic varieties φ : G → V , where each nonempty φ−1(v) is a union of cosets ofH, there exists a unique morphsm σ : G/H → V of algebraic varieties with σ ◦ π = φ. Definition (Homogeneous). An algebraic variety X is called homogeneous, if there exists an algebraic group G, a group action G × V → V and a closed subgroup H ≤ G with X =∼ G/H. Then X homogeneous ⇔ G acts transitively on X. Example. • Let V be a 4-dimensional vector space. Fix (U, W ) ∈ F (V ; 1, 2) with basis {v1} ⊂ {v1, v2} ⊂ {v1, . . . , v4}. Then the stabilizer is

∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ H =∼   F (V ; 1, 2) =∼ Gl(4,K)/H 0 0 ∗ ∗ 0 0 ∗ ∗

• F (V ; 1, . . . , n) =∼ Gl(n, K)/T (n, K), Definition (Borel subgroup). Let G be an algebraic group. Then a Borel sub- group of G is a maximal connected solvable subgroup of G.

5 • all Borel subgroups are conjuagte • all conjugates of a Borel subgroup are Borel subgroups • G/B is a complete variety • H a closed connected subgroup of G. Then H is a Borel subgroup of G if and only if H contains no proper subgroups H0 with G/H0 a projective variety.

Example. T (n, K) is a Borel subgroup of Gl(n, K), because T (n, K) is closed, solvable and connected. With the Lie-Kolchin theorem (if H is a connected solvable subroup of Gl(n, K), then H is conjugate to a subgroup of T (n, K)) it is maximal. So complete flags are complete varieties and the Borel subgroups of Gl(V ) are the stabilizers of complete flags. For a closed subgroup P it is: G/P projective variety ⇔ P contains a Borel subgroup. Definition (Parabolic). A closed subgroup P of an algebraic group G is parabolic, if G/P is a projective variety.

Every stabilizer of a flag is a parabolic subgroup. Proposition. Every parabolic subgroup of Gl(n, K) is the stabilizer of some flag. Proof. Let P be a parabolic subgroup. Then P contains a Borel subgroup. Wlog P ⊂ T (n, K). Then the elements of P have block matrix form with blocks of Pi size mi, i = 1..r. Consider the sequence ni = j=1 mi. Since Gl(V ) acts transitively on F (V ; n1, . . . , nr), there exists a basis and an element {Vni }, such that P is the stabilizer of {Vni }. So the parabolic subroups of Gl(V ) are the subgroups, that stabilizes some flag.

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