Диссертация (1136188), страница 62
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The resulting Newton–Okounkov polytopes were computed in[Ki14, Proposition 4.1]. Regardless of whether s1 corresponds to the shorter or the longer root,these polytopes have 11 vertices (for a strictly dominant weight) while Feigin–Fourier–Littelmann–Vinberg polytopes (as well as string polytopes) for Sp4 have 12 vertices. In particular, the formerare not combinatorially equivalent to the latter.Note that the string polytopes for the decompositionw0 = (s1 )(s2 s1 s2 )(. . .)(sn sn−1 .
. . s2 s1 s2 . . . sn−1 sn ),(Sp)where s1 corresponds to the longer root, coincides (after a unimodular change of coordinates) withthe symplectic Gelfand–Zetlin polytopes by [L, Corollary 6.3]. The latter were exhibited in [O] asthe Newton–Okounkov bodies of the symplectic flag variety Sp2n /B for the lowest term valuationassociated with the B-invariant flag of (not translated) Schubert subvarieties corresponding to theinitial subwords of w0 :Xid ⊂ Xw0 w−1 ⊂ .
. . ⊂ Xw0 w−1 ⊂ Sp2n /B,1d−1where d = n2 = dim Sp2n /B.Finally, note that string polytopes for any connected reductive group G and any reduced decomposition w0 were obtained in [Ka] as the Newton–Okounkov bodies of the complete flag varietyG/B for the highest term valuation associated with the B-invariant flag of Schubert subvarieties:Xid ⊂ Xwd−1 ⊂ . .
. ⊂ Xw1 ⊂ G/B.Here d denotes the dimension of G/B (and the length of w0 ). Note that for G = GLn and w0as in Section 2.1, the string polytope coincides with the Gelfand–Zetlin polytope in type A by [L,Corollary 5.2]. While the highest term valuation comes naturally when dealing with crystal basesand string polytopes the lowest term valuation is more natural from a geometric viewpoint sinceit can be interpreted using the order of the pole of a rational function along a hypersurface.3. Proof of Theorem 2.1We first formulate and prove simple general results about Newton–Okounkov bodies and recallclassical facts about divisors on Schubert varieties.
Then we prove Theorem 2.1.3.1. PreliminariesWe will need the following two simple lemmas on Newton–Okounkov convex bodies.Lemma 3.1. Let X be a variety, L a line bundle on X, and v a valuation on C(X). If D is aneffective divisor on X, then∆v (X, L) ⊂ ∆v (X, L ⊗ O(D)).VALENTINA KIRITCHENKOProof. Since D is effective, 1 ∈ H 0 (X, O(D)).
The lemma follows directly from the definition ofNewton–Okounkov bodies since for any l ∈ N we have the inclusion i : H 0 (X, L⊗l ) ⊂ H 0 (X, (L ⊗O(D))⊗l ) given by i(s) = s ⊗ 1.The lemma below is a partial case of [LM, Theorem 4.24]. We provide a short proof for the reader’sconvenience.Lemma 3.2. Let X ⊂ PN be a projective variety of dimension d, and Y• = ({x0 } = Yd ⊂ . . . ⊂Y1 ⊂ Y0 = X) a complete flag of subvarieties at a smooth point x0 ∈ X. Consider a valuation von C(X) associated with the flag Y• , and the corresponding coordinates a1 , . .
. , ad on Rd . Let v1be the restriction of the valuation v to C(Y1 ). Denote by L the restriction of the dual tautologicalbundle OPN (1) to X. Then we have∆v1 (Y1 , L|Y1 ) = ∆v (X, L) ∩ {a1 = 0}.Proof. It is well-known that the natural restriction map H 0 (PN , OPN (l)) → H 0 (X, L⊗l ) is surjective for sufficiently large l. Similarly, the map H 0 (PN , OPN (l)) → H 0 (Y1 , L⊗l |Y1 ) is surjective.Hence, the map H 0 (X, L⊗l ) → H 0 (Y1 , L⊗l |Y1 ) is surjective, and ∆v1 (Y1 , L|Y1 ) ⊂ ∆v (X, L).
For asection s ∈ H 0 (X, L⊗l ), denote by s̄ its restriction to Y1 . Then s̄ ̸= 0 iff v(s) ∈ {a1 = 0}. Hence,∆v1 (Y1 , L|Y1 ) = ∆v (X, L) ∩ {a1 = 0} as desired.We will also use the classical Chevalley formula [B, Proposition 1.4.3] and the description of Cartierdivisors on Schubert varieties [B, Proposition 2.2.8]. When applied to Xw from (∗) and Lλ thesepropositions immediately yield the followingLemma 3.3. Let w = (si . . .
s1 )(sn−j+1 . . . s1 ) . . . (sn−1 . . . s1 ) where i + j ≤ n. Then the Picardgroup of Xw is spanned by the classes of Xws where s runs through transpositions s1 , s2 . . . , sj−1 ;(j j + 1), (j j + 2),. . . , (j i + j) and (j − 1 i + j + 1), (j − 1 i + j + 2),. . . , (j − 1 n). In particular,Lλ |Xw =j−1⊗O(Xwsl )λl −λl+1 ⊗l=1i⊗O(Xw(jλj −λl+j⊗l+j) )l=1⊗n⊗O(Xw(j−1 l) )λj−1 −λl .l=i+j+1Remark 3.4. Lemma 3.3 implies the following important property of the decomposition w0 . Forevery k ≤ d, the Schubert subvariety Xwk is a Cartier divisor on Xwk−1 . This property is usedin the proof below. It would be interesting to find decompositions with this property for otherreductive groups (decomposition (Sp) for Spn does not have this property).Moreover, it is easy to check that all Xwk are smooth by [M, Theorem 3.7.5] but this is not usedin the proof.3.2.
Proof of Theorem 2.1We will prove by induction the following more general statement. Put Yk := w0 wk−1 Xwk ,and let vk be the restriction of the valuation v to C(Yk ) ≃ C(yk+1 , . . . , yd ) (see Remark2.5). We will also use an alternative labeling of coordinates in Rd , namely, (a1 , a2 , . . . , ad ) =(u1n−1 ; u2n−2 , u1n−2 ; .
. . ; u1n−1 , un−2, . . . , u11 ). Let Fk (λ) be the face of F F LV (λ) given by equations1lum = 0 for all pairs (l, m) such that either m > j, or m = j and l ≥ i. Here k and (i, j) are relatedvia the above identification of coordinates ak and uij , i.e., ak = uij .NEWTON–OKOUNKOV POLYTOPES OF FLAG VARIETIESTheorem 3.5. The Newton–Okounkov convex body ∆vk (Yk , Lλ |Yk ) coincides with the face Fk (λ).In particular, this theorem reduces to Theorem 2.1 when k = 0 (we put F0 (λ) = F F LV (λ)). Themain idea of the proof is to identify the slices of ∆vk−1 (Yk−1 , Lλ |Yk−1 ) by hyperplanes {ak = const}with Fk (µ) for suitable µ. We will need a convex-geometric lemma for slices of Fk−1 (λ) and a similaralgebro-geometric lemma for ∆vk−1 (Yk−1 , Lλ |Yk−1 ).Lemma 3.6.
There exists a path of dominant weights µ(t) such that(t − λi+j )ek + Fk (µ(t)) = Fk−1 (λ) ∩ {ak = t − λi+j }.for all t ∈ [λi+j , λj ]. Here ek denotes the k-th basis vector in Rd . In particular,Fk−1 (λ) = conv{(t − λi+j )ek + Fk (µ(t)) | λi+j ≤ t ≤ λj }.Proof. Define µ(t) = (µ1 (t), . . . , µn (t)) as follows{max{λl , t}µl (t) =λlif j < l ≤ i + jotherwiseIn particular, λ = µ(λi+j ), and every µl (t) is a piecewise linear concave function of t. The lemmanow follows immediately from the definitions of Fk (λ) and F F LV (λ).In particular, Fk−1 (λ) fibers over the segment [0, λj − λi+j ], and the fiber polytope is analogousto Fk (λ) for strictly dominant λ.Lemma 3.7.
Take µ(t) as in the proof of Lemma 3.6. Then(t − λi+j )ek + ∆vk (Yk , Lµ(t) |Yk ) ⊂ ∆vk−1 (Yk−1 , Lλ |Yk−1 ) ∩ {ak = t − λi+j }for all integer t ∈ [λi+j , λj ]. In particular,conv{(t − λi+j )ek + ∆vk (Yk , Lµ(t) |Yk ) | λi+j ≤ t ≤ λj , t ∈ Z} ⊂ ∆vk−1 (Yk−1 , Lλ |Yk−1 ).Proof. By definition, Yk and Yk−1 are translates of the Schubert varieties Xwk and Xwk−1 , respectively, where wk = (si−1 .
. . s1 )(sn−j+1 . . . s1 ) . . . (sn−1 . . . s1 ) and wk−1 = si wk . Put τ = t − λi+j .It is easy to check using Lemma 3.3 thatLλ |Yk−1 ⊗ O(−τ Yk ) = Lµ(t) |Yk−1 ⊗ O(τ (si Yk − Yk )) ⊗ E(τ )for an effective Cartier divisor E(τ ) on Yk−1 . Indeed, E(τ ) = L(λ−µ(t)) |Yk−1 ⊗ O(−τ si Yk ) is atranslate of the following divisor on Xwk−1 :i−1⊗O(Xw(jmax{0,t−λl+j }.l+j) )l=1Note that ∆vk−1 (Yk−1 , Lµ(t) |Yk−1 ⊗O(τ (si Yk −Yk ))) = τ ek +∆vk−1 (Yk−1 , Lµ(t) |Yk−1 ) since si Yk −Ykis the divisor of the rational function yk . Applying Lemma 3.1 to Yk−1 , Lµ(t) |Yk−1 ⊗O(τ (si Yk −Yk ))and E(τ ) we getτ ek + ∆vk−1 (Yk−1 , Lµ(t) |Yk−1 ) ⊂ ∆vk−1 (Yk−1 , Lλ |Yk−1 ⊗ O(−τ Yk )).Intersecting both sides with the hyperplane {ak = τ } yieldsτ ek + ∆vk−1 (Yk−1 , Lµ(t) |Yk−1 ) ∩ {ak = 0} ⊂ ∆vk−1 (Yk−1 , Lλ |Yk−1 ⊗ O(−τ Yk )) ∩ {ak = τ }.Since Lµ(t) is semiample we can apply Lemma 3.2 and get that∆vk (Yk , Lµ(t) |Yk ) = ∆vk−1 (Yk−1 , Lµ(t) |Yk−1 ) ∩ {ak = 0}.It follows thatτ ek + ∆vk (Yk , Lµ(t) |Yk ) ⊂ ∆vk−1 (Yk−1 , Lλ |Yk−1 ⊗ O(−τ Yk )) ∩ {ak = τ }.It remains to note that ∆vk−1 (Yk−1 , Lλ |Yk−1 ⊗ O(−τ Yk )) ⊂ ∆vk−1 (Yk−1 , Lλ |Yk−1 ) by Lemma 3.1.We are now ready to prove Theorem 3.5.VALENTINA KIRITCHENKOProof of Theorem 3.5.
Let us first prove that Fk (λ) ⊂ ∆vk (Yk , Lλ |Yk ) for all dominant λ by backward induction on k. For k = d, we have that both convex bodies coincide with the origin in Rd .Suppose the inclusion holds for k. We now prove it for k − 1. By Lemma 3.6Fk−1 (λ) = conv{(t − λi+j )ek + Fk (µ(t)) | λi+j ≤ t ≤ λj }.Moreover, when taking the convex hull it is enough to consider only integer values of t, since µ(t)is linear at all non-integer points.
Using the induction hypothesis Fk (µ(t)) ⊂ ∆vk (Yk , Lµ(t) |Yk ) weget thatFk−1 (λ) ⊂ conv{(t − λi+j )ek + ∆vk (Yk , Lµ(t) |Yk ) | λi+j ≤ t ≤ λj , t ∈ Z}.Hence, Fk−1 (λ) ⊂ ∆vk−1 (Yk−1 , Lλ |Yk−1 ) by Lemma 3.7.Finally, for k = 0 we get F0 (λ) ⊂ ∆v (GLn /B, Lλ ). Since both convex bodies have the samevolume they must coincide. Here we use that by Theorem 4.3 the volume of F0 (λ) = F F LV (λ)coincides with the volume of the Gelfand–Zetlin polytope GZ(λ). Hence, inclusions Fk (λ) ⊂∆vk (Yk , Lλ |Yk ) are equalities for all k.Remark 3.8. Results of Section 4 (see Theorem 4.3 and Remark 4.1) imply that the number ofinteger points in Fk (λ) (and hence, in the Newton–Okounkov polytope ∆vk (Yk , Lλ |Yk )) is equal tothe dimension of the Demazure module H 0 (Yk , Lλ |Yk ) for all k = 0, . .
. , d and dominant λ.To illustrate the proof of Theorem 3.5 consider the simplest meaningful example.Example 3.9. Let k = d − 1, i.e., wk = s1 and wk−1 = s2 s1 . Then Yk−1 = P̂2 is the blow up of P2at one point, and Yk = P1 is embedded into Yk−1 as one of the fibers of the P1 -bundle P̂2 → P1 .The Picard group of P̂2 is spanned by O(Yk ) and O(E) where E ⊂ P̂2 is the exceptional divisor.Note that O(E)a ⊗ O(Yk )b is semiample iff 0 ≤ a ≤ b. We haveLλ |Yk−1 = O(E)λ1 −λ2 ⊗ O(Yk )λ1 −λ3 .Hence, the line bundle Lλ |Yk−1 ⊗ O(−(t − λ3 )Yk )) is no longer semiample if λ2 < t ≤ λ1 .
However, it has the same global sections (modulo multiplication by ykt−λ3 ) as the semiample bundleLµ(t) = O(E)λ1 −t ⊗ O(Yk )λ1 −t . Hence, Lµ(t) can be used instead of Lλ |Yk−1 ⊗ O(−(t − λ3 )Yk ))when computing ∆vk−1 (Lλ |Yk−1 , Yk−1 ). Figure 2 shows the Newton–Okounkov polygons of Lλ |Yk−1(trapezoid) and Lµ(t) |Yk−1 (triangle), which are just Newton polygons since Yk−1 is toric.4.
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