Flag Varieties. Matthew Goroff November 2, 2016

Transcription

1 Flag Varieties Matthew Goroff November 2, Grassmannian Variety Definition 1.1: Let V be a k-vector space of dimension n. The Grassmannian Grpr, V q is the set of r-dimensional subspaces of V. It is clear that Grp1, V q is equivalent to PpV q, and thus is a projective variety. We will show that all Grassmannians are in fact projective varieties. To do this we will use the Plücker embedding: rľpv φ: Grpr, V q Ñ P q where tu 1,..., u r u is a basis for U. U ÞÑ kpu 1 ^ u 2 ^ ^ u r q Proposition 1.2: Grpr, V q is a projective variety. Proof: First we show the Plücker embedding φ is well-defined. If tu 1 ^ ^ u r u, tu 1 1 ^ ^ u1 ru are two basis for U, then there exists a change of basis matrix M relating these two, and we have u 1 ^ ^ u r detpmqpu 1 1 ^ ^ u1 rq. These are equal in projective space, so φ is well-defined. Now we show that the Plücker embedding is in fact an embedding. Let W P Grpr, V q with basis tw 1,..., w r u, and set w w 1 ^ ^ w r. Define the map ϕ w : V Ñ Ź r`1 pv q by ϕ w pvq v ^ w. It is clear that ϕ w is a linear map, and that W Ď kerpϕ w q. To show the reverse inclusion we extend to the basis tw 1,..., w n u of V and consider v ř n i1 α iw i. v ^ w α i w i ^ w i1 ir`1 ir`1 ir`1 α i w i ^ w α i w i ^ w 1 ^ ^ w r p 1q r α i w 1 ^ ^ w r ^ w i As each w 1 ^ ^ w r ^ w i is unique for each i ą r, if v ^ w 0 then α i 0 for all i ą r. This implies that if v P kerpϕ w q then v ř r i1 α iw i, so v P W. So we have that W kerpϕ w q. Now let U P Grpr, V q with basis tu 1,..., u r u, set u u 1 ^ ^ u r. If φpw q φpuq then kw ku, thus kerpϕ w q kerpϕ u q. So from above we have that W U, making φ an embedding. 1

2 Aside: Notice that if we fix a basis tv 1,..., v n u for V, then we get a basis for Źr pv q. So given a W P Grpr, V q, we can use the Plücker embedding to find the coordinates of W in P N. Example 1.3: Let V have basis tv 1, v 2, v 3 u, and let W P Grp2, V q. Then W has some basis tw 1, w 2 u which we can write in terms of the basis of V : w 1 a 1 v 1 ` a 2 v 2 ` a 3 v 3 and w 2 b 1 v 1 ` b 2 v 2 ` b 3 v 3. So: w 1 ^ w 2 pa 1 v 1 ` a 2 v 2 ` a 3 v 3 q ^ pb 1 v 1 ` b 2 v 2 ` b 3 v 3 q pa 1 b 2 a 2 b 1 qv 1 ^ v 2 ` pa 1 b 3 a 3 b 1 qv 1 ^ v 3 ` pa 2 b 3 a 3 b 2 qv 2 ^ v 3 So the Plücker coordinates are: pa 1 b 2 a 2 b 1, a 1 b 3 a 3 b 1, a 2 b 3 a 3 b 2 q. If we let w w 1 ^ w 2 then we can express the map ϕ w in matrix form: `a2 b 3 a 3 b 2 pa 1 b 3 a 3 b 1 q a 1 b 2 a 2 b 1 Example 1.4: Let V have basis tv 1, v 2, v 3 u, and let W P Grp1, V q. Then W has some basis twu which we can write as w a 1 v 1 ` a 2 v 2 ` a 3 v 3. With Plücker coordinates: pa 1, a 2, a 3 q. We can express the map ϕ w in matrix form: a 1 0 a2 a 3 0 a 1 0 a 3 a 2 Now we prove that the image of the Plücker embedding is Zariski closed. Consider v P V and w P Źr pv q. We say that v divides w if w v ^ u for some u P Źr 1 pv q. Notice that v divides w implies that v ^ w 0 as then w must be the wedge product of some scale of v. We will show that v ^ w 0 implies that v divides w. Choose a basis for V that includes v, and write w in the induced basis on Źr pv q. Now when we take v ^ w, we see that each term either has two v s or is a basis element of Źr`1 pv q. As the basis elements cannot cancel with each other, if v ^ w 0 then the only terms of w that occur are those that contain a v. Thus v divides w. Now consider kerpϕ w q. We have just shown that if v P kerpϕ w q then there exists some u P Źr 1 pv q such that w v ^ u. So let w 1,..., w t P kerpϕ w q be linearly independent vectors in V. Then we have that w w 1 ^ ^ w t ^ u for some u P Źr t pv q. Thus t ď r, so dimpkerpϕ w qq ď r. So rankpϕ w q ě n r. But also notice that if w w 1 ^ ^ w r (i.e. w is a decomposable element) then by above we have that rankpϕ w q n r. This gives us that w is decomposable if and only if rankpϕ w q ď n r. Recall that the image we are concerned with are exactly the scalers of decomposable elements. Thus we have shown that an element kw is in the image if and only if rankpϕ w q ď n r. This is a polynomial condition on the matrix of ϕ w, and we know that the matrix is formed of (up to sign) the Plücker coordinates of w. Thus whether or not an element is in the image is a polynomial condition on it s coordinates, meaning that the image is Zariski closed. 2

3 2. Flag Varieties Definition 2.1: Let V be a finite dimensional vector space. A flag is a nested sequence of subspaces of V : V 1 Ă V 2 Ă Ă V r The signature of a flag is the set of dimensions of the subspaces: pdimpv 1 q,..., dimpv r qq. Definition 2.2: Let V be a finite dimensional vector space. A flag variety is the set of all flags of a particular signature. We write: FpV ; n 1,..., n r q tv 1 Ă Ă V r dimpv i q n i u A flag variety of the form FpV ; 1,..., nq is called a complete flag variety. Otherwise we say that FpV ; n 1,..., n r q is a partial flag variety. Proposition 2.3: Let V be a vector space and 0 ă n 1 ă ă n r ď n, then FpV ; n 1,..., n r q is a projective variety. Proof: There is an obvious embedding: Ψ: FpV ; n 1,..., n r q Ñ Grpn 1, V q ˆ ˆ Grpn r, V q We will show that the image is Zariski closed. Let π ij : Grpn 1, V q ˆ ˆ Grpn r, V q Ñ Grpn i, V q ˆ Grpn j, V q be the projection for i ă j. Notice that: ΨpFpV ; n 1,..., n r qq č π 1 ij pψpfpv ; n i, n j qqq iăj So it is sufficient to show that for all i ă j, ΨpFpV ; r, sqq is closed. Let tv 1,..., v n u be a basis for V and let pu, W q P Grpr, V q ˆ Grps, V q. Now let tu 1,..., u r u be a basis for U and tw 1,..., w s u be a basis for W. Now set u u 1 ^ ^ u r and w w 1 ^ ^ w s. We have the maps ϕ u and ϕ w from before, and we can construct the map: ϕ u ϕ w : V Ñ r`1 ľ ľs`1 pv q pv q From before we know that kerpϕ u q U and kerpϕ w q W, so it is clear that kerpϕ u ϕ w q U XW. So we have that: rankpϕ u ϕ w q dimpv q dimpkerpϕ u ϕ w qq dimpv q dimpu X W q ě dimpv q dimpuq n r This implies that U Ă W if and only if rankpϕ u ϕ w q n r if and only if rankpϕ u ϕ w q ď n r. As before, we can represent ϕ u ϕ w by a matrix with respect to a basis, and see that the entries are Plücker coordinates (up to sign) of u and w. rankpϕ u ϕ w q ď n r is a polynomial condition on this matrix, and thus on the coordinate of u and v, thus we have that U Ă W exactly at the zeros of a set of polynomials. So ΨpFpV ; r, sqq is Zariski closed. Example 2.4: Let V be a 3-dimensional vector space with basis tv 1, v 2, v 3 u. Let pu Ă W q P FpV ; 1, 2q with tuu a basis for U and tw 1, w 2 u a basis for W. We see that: U Ă W ðñ kerpϕ u ϕ w q U 3

4 So if we let t t 1 v 1 ` t 2 v 2 ` t 3 v 3 then we have that: U Ă W ðñ t 1 b 23 t 2 b 13 ` t 3 b 12 0 A homogeneous polynomial. 3. Algebraic Groups Definition 3.1: Let G be a group with the structure of an affine algebraic variety such that the multiplication map G ˆ G Ñ G and the inverse map G Ñ G are regular maps of algebraic varieties. Then G is a linear algebraic group. Example 3.2: Glpn, kq is an algebraic group. It is the complement of the zero set of the determinant, a single polynomial. Thus it is an affine variety. The formulas for matrix multiplication and inverse can easily be seen as polynomial equations (as det 0). Definition 3.3: If G is an algebraic group, then we call any maximal connected solvable closed subgroup a Borel subgroup. Example 3.4: Consider T pn, kq, the set of upper-triangular invertible matrices. T pn, kq is the zero set of the polynomials x ij 0 for i ą j in Glpn, kq. So it is also an algebraic group, thus it is Zariski closed. T pn, kq is also connected and solvable, so if it were the maximal such subgroup in Glpn, kq then it would be a Borel subgroup. The Lie-Kolchin theorem tells us that if H is a connected solvable subgroup of Glpn, kq, then H is conjugate to a subgroup of T pn, kq. Thus T pn, kq is maximal among connected solvable subgroups of Glpn, kq so it is a Borel subgroup. Definition 3.5: Let G be an algebraic group and V a variety. we say that G acts on V if there is a group action G ˆ V Ñ V that is also a regular map of algebraic varieties. Just as we can study orbits and stabilizers of group actions, we can study them on algebraic group actions as well. Consider the group action of Glpn, kq where A P Glpn, kq Proposition 3.6: Glpn, kq acts transitively on flag varieties. Proof: Let pu n1 Ă Ă U nr q, pw n1 Ă Ă W nr q P FpV ; n 1,..., n r q. Choose basis tu 1,..., u n u and tw 1,..., w n u of V such that tu 1,..., u ni u is a basis for U ni and tw 1,..., w ni u is a basis for W ni. We can pick an A P Glpn, kq such that Apu i q w i, a change of basis matrix. Proposition 3.7: T pn, kq stabilizes complete flags. 4

5 Definition 3.8: Let G be an algebraic group. If P is a closed subgroup of G where G{P is a projective variety then we call P a parabolic subgroup. Notice that P contains a Borel subgroup. Proposition 3.9: Every parabolic subgroup of Glpn, kq is the stabilizer of some flag. Proof: Let P be a parabolic subgroup. A calculation shows that P is in block matrix form with r blocks. Let FpV ; n 1,..., n r q be the flag variety with subscripts the same as for P. Let te 1,..., e n u be a basis for V, and set F to be the flag pv n1 Ă Ă V nr q where V ni is the span of the first n i of the e s. It is clear that P stabilizes F. References: 1) Letz, Janina Flag Varieties ) Morandi, Patrick Algebraic Groups, Grassmannians, and Flag Varieties