Flag Varieties
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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 ) = fU ⊂ V jr − dimensional subspaceg 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 fu1; : : : ; urg a basis of U. 1. Φ well-defined 0 0 0 fu1; : : : ; urg and fu1; : : : ; urg 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 2 Gr(r; V ) with basis fu1; : : : ; urg and fw1; : : : ; wrg 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 ) , 9x 2 K : u = xw , 9x 2 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 fw1; : : : ; wng a basis of V . P Let v 2 ker(φw), v = aiwi. Then n X 0 = v ^ w = aiwi ^ w ) ai = 0 i ≥ r + 1 ) v 2 W i=r+1 1 Vr 3. Φ is closed in P( (V )) (a) v 2 V; w 2 Vr(V ). Then v divides w , v ^ w = 0 in Vr+1(V ) One direction is clear. For the other choose a basis fv1; : : : ; vng of V with v = v . Then w = P a v ^ · · · ^ v . Then for 1 1≤i1<···<ir ≤n i1:::ir i1 ir 0 = v ^ w it is ai1:::ir = 0 if i1 6= 1. So v divides w. Vr (b) w 2 (V ). Let w1; : : : ; wt be a basis of ker(φw). So w = w1 ^ · · · ^ Vr−t wt ^ u for u 2 . 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 2 (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 2 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 8k b = k ji1:::ir+1 (−1)ka i = j i1:::^ik:::ir+1 k n for f : f1 ::: r+1 ! f(i1; : : : ; ir+1)j1 ≤ i1 < ··· < ir+1g a bijection of sets. So Gr(r; V ) =∼ V (f(n + r − 1)-minor of B = 0g). 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 fv1; : : : ; v4g. • Let W 2 Gr(1;V ) with basis fwg. X w = aivi Matrix representation of φw: 0 1 a2 −a1 0 0 Ba3 0 −a1 0 C B C Ba4 0 0 −a1C B C B 0 a3 −a2 0 C B C @ 0 a4 0 −a2A 0 0 a4 −a3 3 Gr(1;V ) =∼ V (f0g) =∼ P 2 • Let W 2 Gr(2;V ) with basis fw1; w2g. X X X w1 = aivi w2 = bivi w1 ^ w2 = (aibj − ajbi) vi ^ vj i<j | {z } aij Matrix representation of φw1^w2 : 0 1 a23 −a13 a12 0 Ba24 −a14 0 a12C B C @a34 0 −a14 a13A 0 a34 −a24 a23 So there are 16 equations. V2 Another way: Use, that for u 2 (V ) it is 9ui 2 V for u = u1 ^ u2 , u ^ u = 0. So ∼ Gr(2;V ) = V (X12X34 − X13X24 + X14X23) • Let W 2 Gr(3;V ) with basis fw1; w2; w3g. X X X w1 = aivi w2 = bivi w3 = civi X w1^w2^w3 = (aibjck + ajbkci + akbicj − akbjci − ajbick − aibkcj) vi^vj^vk i<j<k | {z } aijk Matrix representation of φw1^w2^w3 : a234 −a134 a124 −a123 3 Gr(3;V ) =∼ V (fg) =∼ P 2 Flag variety Definition (Flag variety). F (V ; n1; : : : ; nr) = f0 ⊂ V1 ⊂ · · · ⊂ Vr ⊂ V j dim(Vi) = nig 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 ) fVig 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<j 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 ) 2 Gr(r; V ) × Gr(s; V ) and fu1; : : : ; ur) and fw1; : : : ; wsg 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 ) 2 F (V ; 1; 2), fug P P and fw1; w2g basis of U and W . u = aivi and w1 ^ w2 = i<j aijvi ^ vj. So X u ^ w1 ^ w2 = (aiajk − ajaik + akaij)vi ^ vj ^ vk i<j<k ∼ Then U ⊂ W , u ^ w1 ^ w2 = 0. So F (V ; 1; 2) = V (fX1X23 − X2X13 + 9 X3X12;::: g) ⊂ P . Definition (Complete). A flag variety of the form F (V ; 1; : : : ; n) is called com- plete. 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 nV (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 (fXijji < jg) \ 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 fUni g; fWni g 2 F (V ; n1; : : : ; nr). Then choose basis fu1; : : : ; ung and fw1; : : : ; wng of V , such that fu1; : : : ; uni g a basis of Uni and fw1; : : : ; wni g basis of Wni . Then define A 2 Gl(V ) as A(ui) = wi. Let fVig 2 F (V ; 1; : : : ; n) and fv1; : : : ; vng a basis corresponding to fVig. ∼ Then Gl(V ) = Gl(n; K) by the basis and the stabilizer of fVig 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 ) 2 F (V ; 1; 2) with basis fv1g ⊂ fv1; v2g ⊂ fv1; : : : ; v4g. Then the stabilizer is 0∗ ∗ ∗ ∗1 B0 ∗ ∗ ∗C H =∼ B C F (V ; 1; 2) =∼ Gl(4;K)=H @0 0 ∗ ∗A 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.