Диссертация (1136188), страница 61

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 61)

Label coordinates in Rd corresponding to (y1 , . . . , yd )by (u1n−1 ; u2n−2 , u1n−2 ; . . . ; u1n−1 , un−2, . . . , u11 ). Arrange the coordinates into the table1λ1λ2u11u21λ3u12...u2n−2.....λn......un−21u1n−1(F F LV ).un−11un−22The polytope F F LV (λ) is defined by inequalities ulm ≥ 0 and∑ulm ≤ λi − λj(l,m)∈Dfor all Dyck paths going from λi to λj in table (F F LV ) where 1 ≤ i < j ≤ n (see [FFL] for moredetails).Example 2.2. (a) For n = 3, there are six inequalities0 ≤ u11 ≤ λ1 − λ2 ;0 ≤ u12 ≤ λ2 − λ3 ;0 ≤ u21 ;u11 + u21 + u12 ≤ λ1 − λ3 .In this case, there is a unimodular change of coordinates that maps F F LV (λ) to the Gelfand–Zetlinpolytope GZ(λ) (see Section 4 for a definition of GZ(λ)).(b) For n = 4, there are 13 inequalities0 ≤ u11 ≤ λ1 − λ2 ;0 ≤ u12 ≤ λ2 − λ3 ;u11 + u21 + u12 ≤ λ1 − λ3 ;u11 + u21 + u12 + u22 + u13 ≤ λ1 − λ4 ;0 ≤ u13 ≤ λ3 − λ4 ;0 ≤ u21 , u22 , u31 ;u12 + u22 + u13 ≤ λ2 − λ4 ;u11 + u21 + u31 + u22 + u13 ≤ λ1 − λ4 .In this case, F F LV (λ) and GZ(λ) are combinatorially different whenever λ is strictly dominantbecause they have different number of facets (cf.

[Fo, Proposition 2.1.1]).2.2. CoordinatesWe now introduce coordinates on the open Schubert cell in GLn /B that are compatible with theflag (∗). These coordinates seem to be natural from a geometric viewpoint and will be used tocompute by hand some examples in the end of this section. However, they are not needed for theproof of the main result.VALENTINA KIRITCHENKOTo motivate the definition consider first the Bott–Samelson variety Xw0 .

Its points are collecitions of d subspaces {Vji ⊂ Cn | i + j ≤ n, i, j > 0} such that dim Vji = i, and Vji , Vj+1⊂ Vji+1i+1where we put Vn−i := F i+1 . Incidence relations between subspaces Vji can be organized into thefollowing table (similar to the Gelfand–Zetlin table).V11V12V21...2Vn−2........V1n−2.V1n−1V2n−21Vn−1F1F2···F n−2Fn−1where the notationUVWmeans U, V ⊂ W .Collections of spaces (Vji ⊂ Cn | i + j ≤ n, i, j ≥ 1) appear naturally when we start from thefixed flag F • and apply d one parameter deformations to get the moving flag M • := (V11 ⊂ V12 ⊂. . . ⊂ V1n−1 ⊂ Cn ). The deformations are encoded by the word w0 as follows. The elementarytransposition si corresponds to P1 -family of complete flags that differ only in the i-th subspace.1To go from F • to M • we first move F 1 inside F 2 and get the flag (Vn−1⊂ F 2 ⊂ . .

. ⊂ F n−1 ),3n−123211), third we move Vn−1second we move F inside F and get (Vn−1 ⊂ Vn−2 ⊂ F ⊂ . . . ⊂ F12inside Vn−2 to get Vn−2 and so on.Example 2.3. Let n = 4. Below is the sequence of intermediate flags between F • and M • .1213F• →(V31 ⊂ F 2 ⊂ F 3 ) →(V31 ⊂ V22 ⊂ F 3 ) →(V21 ⊂ V22 ⊂ F 3 ) →ssss21(V21 ⊂ V22 ⊂ V13 ) →(V21 ⊂ V12 ⊂ V13 ) →M•ssRemark 2.4.

The word w0 is the same (after switching si and sn−i ) as the word used in [V, 2.2] toencode the path from the fixed flag to the moving flag in order to establish a geometric Littlewood–Richardson rule for Grassmannians. According to [V, 3.12] not every reduced decomposition of w0can be used for this purpose which is another manifestation of the special properties of w0 .Note that if F • and M • are in general position (that is, M • lies in the open Schubert cell C withrespect to F • ), then all subspaces Vji are uniquely defined by M • , namely, Vji = F n−j+1 ∩ M i+j−1 .In particular, the natural projectionπw0 : Xw0 → GLn /B;πw0 : (Vji ) 7→ M •is one to one over C.

Fix a basis e1 ,. . . , en in Cn compatible with F • , i.e., F i = ⟨e1 , . . . , ei ⟩(fixing such a basis is equivalent to fixing a maximal torus T ⊂ B, and hence, an action of the Weyl group on flags). Using the word w0 we now introduce natural coordinates−1(x1n−1 ; x2n−2 , x1n−2 ; .

. . ; xn−1, x1n−2 , . . . , x11 ) on C ≃ πw(C). The origin in this coordinate system10•12n−1is the flag w0 F := (w0 F ⊂ w0 F ⊂ . . . ⊂ w0 F). The coordinate xij determines the positioni−1i1of Vj inside the P -family of dimension i subspaces Vji (xij ) such that Vj+1⊂ Vji (xij ) ⊂ Vji+1 . Todefine the coordinate xij on P1 uniquely up to a constant factor it is enough to choose Vji (0) andVji (∞). The following choice seems to be the most natural.NEWTON–OKOUNKOV POLYTOPES OF FLAG VARIETIESPHF 1 LPHF 2 LPHV21 LPHV11 H0LLPHV21 H0LLPHV12 H0LLPHM 1 LPHM 2 LPH<e3 >LFigure 1. Coordinates on flags for n = 3.Since M • and F • are in general position, that is, dim(F n−j ∩ M i+j ) = i, we have inclusions ofpairwise distinct subspaces:Vji = F n−j+1 ∩ M i+j−1i−1Vj+1= F n−j ∩ M i+j−1̸=i+1n−j+1Vj=F∩ M i+jiVj+1= F n−j ∩ M i+ji−1iPut Vji (∞) := Vj+1and Vji (0) := ⟨F n−i−j , en−j+1 ⟩ ∩ M i+j + Vj+1.

Note that ⟨F n−i−j , en−j+1 ⟩ ∩M i+j is the line spanned by a vector en−j+1 + v for some v ∈ F n−i−j since F n−i−j ∩ M i+j = {0}./ F n−j . By construction,It follows that dim Vji (0) = i, and Vji (0) ̸= Vji (∞) because en−j+1 ∈i−1i+1ii11iVj+1 ⊂ Vj (0) ⊂ Vj . Note also that Vj lies in A = P \ {Vj (∞)} when M • and F • are ingeneral position.Remark 2.5. Itisnothardtocheckthatcoordinates(y1 , . .

. , yd ):=n−21(x1n−1 ; x2n−2 , x1n−2 ; . . . ; xn−1,x,...,x)arecompatiblewiththeflag(∗)ofSchubertsub111varieties.Example 2.6. Let n = 3. ThenV11 = ⟨(x11 x12 − x21 )e1 + x11 e2 + e3 ⟩;V21 = ⟨x12 e1 + e2 ⟩;V12 = ⟨x12 e1 + e2 , −x21 e1 + e3 ⟩.Figure 1 depicts projectivizations in P2 of various subspaces involved in this example.2.3.

ExamplesTheorem 2.1 will be proved in the next section. Here we verify it by hand in three simplestexamples.VALENTINA KIRITCHENKOExample 2.7. cf. [An, Section 6.4] Let n = 3, and λ = (2, 1, 0). The flag variety GL3 /B can be∗regarded as a hypersurface in P2 × P2 under the embedding (V11 , V12 ) 7→ V11 × V12 . The line bundleLλ on GL3 /B is the pullback of the dual tautological line bundle O(1) on P8 under the embedding:∗ Segrepλ : GL3 /B ,→ P2 × P2 −→ P8 .Using Example 2.6 we get that in coordinates (y1 , y2 , y3 ) = (x12 , x21 , x11 ) the map pλ takes the formy1 y3 − y2() × y2y3y11 .pλ : (y1 , y2 , y3 ) 7→ 1Hence, H 0 (GL3 /B, Lλ ) has the basis 1, y1 , y2 , y3 , y1 y3 , y2 y3 , y1 y2 y3 − y22 , y12 y3 − y1 y2 . Applyingthe valuation v we get 8 integer points (0, 0, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1), (0, 2, 0),(1, 1, 0), whose convex hull in R3 is given exactly by inequalities of Example 2.2(a).Example 2.8.

Let n = 4, and λ = (1, 1, 0, 0). The line bundle Lλ on GL4 /B is the pullbackof the dual tautological line bundle O(1) on P5 under the natural projection GL4 /B → G(2, 4)composed with the Plücker embedding G(2, 4) ,→ P5 of the Grassmannian. Using Example 2.3 weget that in coordinates (y1 , . . . , y6 ) the plane V12 is spanned by the vectors (y4 y6 + y5 , y4 , 1, 0) and(y2 y6 + y3 , y2 , 0, 1).

Hence, the map pλ has the formpλ : (y1 , . . . , y6 ) 7→ (y2 y5 − y3 y4 : −(y2 y6 + y3 ) : y4 y6 + y5 : −y2 : y4 : 1).The valuation v takes the sections of H 0 (GL4 /B, Lλ ) to 6 integer points in the 4-space {u11 = u13 =0}. In coordinates (u21 , u31 , u12 , u22 ), these points are (0, 1, 1, 0), (0, 1, 0, 0), (0, 0, 0, 1), (1, 0, 0, 0),(0, 0, 1, 0), (0, 0, 0, 0).

Their convex hull in R4 is given exactly by inequalities of Example 2.2(b).Example 2.9. The previous example can be extended to G(3, 6), that is, n = 6 and λ =(1, 1, 1, 0, 0, 0). This is the minimal example when F F LV (λ) and GZ(λ) are not combinatorially equivalent (cf. [Fo, Proposition 2.1.1]). When computing V13 in coordinates (y1 , . . . , y15 ) onecan immediately ignore all monomials that contain y15 , y14 , y13 since they never appear as thelowest order terms. The same holds for y3 , y2 , y1 . If y15 = y14 = y13 = 0, then pλ takes thefollowing simple form:y10 y11 y12 1 0 0pλ : (y4 , . . .

, y12 ) 7→ 3 × 3 minors of  y7 y8 y9 0 1 0 .y4 y5 y6 0 0 1Hence, we have to compute the lowest order terms of all minors of the 3 × 3 matrix formed by thefirst three columns. After rotating this matrix as followsy10y7y4y11y8y5y12y9y6it is easy to see that the lowest order monomials in the minors are in bijective correspondence withthose collections of uij (where 3 ≤ i + j ≤ 6, j ≤ 3) in table (F F LV ) that can not occur in thesame Dyck path. By definition, F F LV (λ) contains an integer point with uij = 1 and ulm = 1 iffno Dyck path passes through both uij and ulm . Hence, the valuation v maps bijectively the minorsto the integer points in F F LV (λ).NEWTON–OKOUNKOV POLYTOPES OF FLAG VARIETIESRemark 2.10. Arguments of Example 2.9 allow one to identify ∆v (GLn /B, Lωi ) with F F LV (ωi )for any fundamental weight ωi of GLn . This might lead to an alternative proof of Theorem 2.1if one uses that ∆v (GLn /B, Lλ ) for λ = k1 ω1 + .

. . + kn−1 ωn−1 contains the Minkowski sumk1 ∆v (GLn /B, Lω1 ) + . . . + kn−1 ∆v (GLn /B, Lωn−1 ).2.4. Symplectic caseA statement analogous to Theorem 2.1 does not hold in type C already in the case of Sp4 . We nowdiscuss this case in more detail. For the rest of this section, X denotes the complete flag varietyfor Sp4 . The flag of translated Schubert subvarieties analogous to (∗) has the forms1 s2 s1 s2 Xid ⊂ s1 s2 s1 Xs2 ⊂ s1 s2 Xs1 s2 ⊂ s1 Xs2 s1 s2 ⊂ X,where s1 , s2 are simple reflections.

Характеристики

Список файлов диссертации