Диссертация (1136188), страница 53

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Note that Dαi (ep(c)+w0 λ tσi (z) ) = ep(c)+w0 λ TΠc (tσi (z) ) becausePλ is λ-balanced. Hence,⎛⎞ep(c)tσi (z) ⎠D α i ⎝e w 0 λc∈πi (S)∩Zd−di= ew 0 λ⎛z∈Sc ∩Zdiep(c) TΠc ⎝⎞tσi (z) ⎠z∈Sc ∩Zdic∈πi (S)∩Zd−diwhere TΠc is the operator defined in Section 2.

By [KST, Proposition 6.10],which is applicable because of hypotheses (1)–(3), we get⎛⎞TΠ c ⎝z∈Sc ∩tσi (z) ⎠ =Z ditσi (z) .z∈Mi (S)c ∩ZdiHence,⎛ep(c) TΠc ⎝⎞tσi (z) ⎠ =z∈Sc ∩Zdic∈πi (S)∩Zd−diep(c)c∈πi (S)∩Zd−ditσi (z)z∈Mi (S)c ∩ZdiFinally, since πi (S) = πi (Mi (S)) by (4) we getc∈πi (S)∩Zd−diep(c)z∈Mi (S)c ∩Zditσi (z) =Mi (S)∩Zdep(x) .�This theorem gives an inductive algorithm for realizing every Demazurecharacter as the exponential sum over the unions of certain faces of Pλ if Pλsatisfies an extra assumption.Definition 10. A λ-balanced parapolytope Pλ ⊂ Rd with the lowest vertex0 is called admissible if dim P ∩ Rdi ≤ 1 for all i = 1, . . .

, r.Geometric mitosis1085Remark 3.5. DDO polytopes of [K13, Section 3] are admissible (see thediscussion at the end of [K13, Section 4.3]). In particular, polytopes GZ λ −aλ and SP λ are admissible, which is easy to check directly.We now discuss the algorithm. Let B ⊂ G be a Borel subgroup, andX = G/B complete flag variety. For an element w ∈ W of the Weyl group,denote by Xw = BwB/B the Schubert variety corresponding to w. We willalso consider the opposite Schubert varieties X w = B − wB/B where B − ⊂ Gdenotes the opposite Borel subgroup.

Note that Schubert cycles [X w0 w ] and[Xw ] coincide in H ∗ (G/B, Z). Recall that with a dominant weight λ of G,one can associate a G-linear line bundle Lλ on the complete flag varietyX = G/B so that H 0 (X, Lλ ) = Vλ∗ as G-modules.+The Demazure B-submodule Vλ,wcan be defined as H 0 (Xw , Lλ |Xw )∗ .−Similarly, Demazure B − -submodule Vλ,wcan be defined as H 0 (Xw , Lλ |X w )∗ .+−wand Vλ,w, respectivelyLet χw (λ) and χ (λ) denote the characters of Vλ,w(they are called Demazure characters). It is easy to check that w0 χw (λ) =χw0 w (λ).

Let sj1 · · · sj be a reduced decomposition of w0 ww0−1 such that(j1 , . . . , j ) is a subword of (i1 , . . . , id ).Corollary 3.6. Let Pλ ⊂ Rd be an admissible λ-balanced parapolytope, andSw ⊂ Pλ the union of all faces produced from the vertex 0 ∈ Pλ by applyingsuccessively the operations Mj , . . . , Mj1 .

Suppose that for every 1 < k ≤ ,the collection of faces Mjk · · · Mj (0) satisfies conditions (3) and (4) of Theorem 3.4. Thenχw0 w (λ) = ew0 λep(x) .x∈Sw ∩ZdProof. By the Demazure character formula [A] we haveχw0 w (λ) = Dαj1 · · · Dαj ew0 λ .We now proceed by induction applying Theorem 3.4 repeatedly to the righthand side. Note that conditions (1) and (2) of this theorem are fulfilled forMik · · · Mi1 (0) for all k < .

Indeed, if a face Γ contains 0 then all faces inMi (Γ) contain 0 because Pλ is admissible, and by Remark 2.4 the mitosisapplied to a union of L-classes produces a union of L-classes.�For G = GLn and GZ λ − aλ , this corollary reduces to [KST, Theorem 5.1]and holds for all w ∈ W . It is easy to check that for G = Sp4 and SP λ , conditions of Corollary 3.6 are also satisfied for all w. More generally, condition1086Valentina Kiritchenko(4) is satisfied for all w if Pλ is a DDO polytope of [K13, Theorem 3.6] (simply by construction of these polytopes).

Condition (3) is trickier to checkas the case of Gelfand-Zetlin polytopes shows (see [KST, Lemma 6.13]).Whenever Corollary 3.6 holds for all w ∈ W , the general results of [KST,Section 2] on polytope rings allow one to model Schubert calculus on G/Bby intersecting faces of Pλ . For GLn and Gelfand–Zetlin polytopes this wasdone in [KST], and the example with Sp4 and SP λ will be considered in thenext section.4. Sp4 exampleWe now apply the results of the preceding section to Sp4 and the symplecticDDO polytope SP λ from Example 2.9. We explain an algebro-geometricmeaning of SP λ and outline applications of Corollary 3.6 to the Schubertcalculus on Sp4 /B.4.1.

Newton–Okounkov convex bodiesRecall briefly the definition of Newton–Okounkov convex bodies in the simplest case (for more details see [KaKh]). Let X be a projective variety ofdimension d, and L a very ample line bundle on X. By fixing a global sections0 of L we can identify the space of global sections H 0 (X, L) with a subspaceof the field of rational functions C(X). Let v : C(X) \ {0} → Zd be a surjective valuation. For instance, one can choose local coordinates x1 , . .

. , xdon X and assign to every polynomial its lowest order term with respectto some ordering on Zd . More geometrically, take a full flag of subvarieties{x0 } = X0 ⊂ X1 ⊂ · · · ⊂ Xd = X at a smooth point x0 ∈ X and assign toevery rational function its (properly defined) orders of vanishing along Xiconsidered as a hypersurface in Xi+1 (see [KaKh, Examples 2.12, 2.13] formore details).The Newton–Okounkov convex body Δv (X, L) is defined as the closurev(s/sk0 )| s ∈ H 0 (X, L⊗k )} ⊂ Zd ⊂ Rd .of the convex hull of the set ∞k=1 {kExplicit description of Δv (X, L) (e.g. by inequalities) is usually a challengingtask.

Sometimes, it is enough to computeΔ1v (X, L) = conv{v(s/s0 ) | s ∈ H 0 (X, L)},that is, the first polytope approximation of Δv (X, L). By [KaKh, Corollary 3.2], we have that Δ1v (X, L) = Δv (X, L) whenever the volume of Δ1v (X, L)times d! coincides with the degree of X embedded into P(H 0 (X, L)∗ ) (thisGeometric mitosis1087argument is used in the proof of Proposition 4.1 below). Note that if X isa toric variety and v is any valuation defined using standard coordinates onthe open torus orbit (C∗ )d ⊂ X, then Δv (X, L) coincides with the classicalNewton (or moment) polytope of X.4.2. DDO polytope as a Newton–Okounkov bodyLet us discuss the algebro-geometric interpretation of the symplectic DDOpolytope.

Recall that α1 denotes the shorter root and α2 denotes the longerone. Let ω1 , ω2 be the corresponding fundamental weights, and λ = λ1 ω1 +λ2 ω2 a dominant weight of Sp4 . We are going to identify SP λ with theNewton-Okounkov polytope of Lλ for a natural geometric valuation v on X.To define the valuation v we introduce coordinates on an open Schubertcell in X. Choose a basis in C4 so that ω := e∗1 ∧ e∗4 + e∗2 ∧ e∗3 is the symplecticform preserved by Sp4 .

Points in X can be identified with isotropic complete⊥flags (V 1 ⊂ V 2 ⊂ V 3 ⊂ C4 ). A flag is isotropic if ω|V 2 = 0 and V 3 = V 1 :={v ∈ C4 | ω(v, u) = 0 ∀u ∈ V 1 }. Taking projectivization we also identifypoints in X with projective partial flags (a = P(V 1 ) ∈ l = P(V 2 )). Fix theflag (a0 , l0 ) ∈ X where a0 = (1 : 0 : 0 : 0) and l0 = a0 , (0 : 1 : 0 : 0), i.e.,(a0 , l0 ) is the fixed point for the upper-triangular Borel subgroup B ⊂ Sp4 .The open Schubert cell in X with respect to (a0 , l0 ) consists of all (a, l) suchthat (a0 , l0 ) and (a, l) are in general position (i.e., a0 ∈/ l, a ∈/ l0 , l 0 ∩ l = ∅etc). The Schubert varieties with respect to (a0 , l0 ) can be described as follows:Xid = {(a0 , l0 )}; Xs1 = {l = l0 }; Xs2 = {a = a0 };Xs1 s2 = {a ∈ l0 };Xs2 s1 = {a0 ∈ l};Xs2 s1 s2 = {a ∈ a⊥0 };Xs1 s2 s1 = {l ∩ l0 = ∅};Xs1 s2 s1 s2 = Xs2 s1 s2 s1 = X.Define coordinates on the open Schubert cell:a = (y + xz : z : −x : 1);l = a, (z + xt : t : 1 : 0).These coordinates are chosen so that the flag {x = y = z = t = 0} ⊂ {x =y = z = 0} ⊂ {x = y = 0} ⊂ {x = 0} ⊂ X coincides with the flag of translated Schubert subvarieties:s1 s2 s1 s2 Xid ⊂ s1 s2 s1 Xs2 ⊂ s1 s2 Xs1 s2 ⊂ s1 Xs2 s1 s2 ⊂ X(after intersecting with the open Schubert cell).

The flag corresponds to thedecomposition w0 = s1 s2 s1 s2 , and the coordinates (x, y, z, t) come naturally1088Valentina Kiritchenkow0 (see [Ka13, Section 2.2] orif one considers the Bott–Samelson variety X[K15, Section 2.2]). Fix the lexicographic ordering on monomials in x, y, z, t,i.e., xk1 y k2 z k3 tk4 xl1 y l2 z l3 tl4 iff there exists j ≤ 4 such that ki = li for i < jand kj > lj . Let v denote the lowest order term valuation on C(X) associatedwith the flag and ordering (cf. [An, Section 6.4], [Ka13, Remark 2.3], wherev is denoted by v w0 ), and Δv (X, Lλ ) ⊂ R4 the Newton–Okounkov convexbody corresponding to X, Lλ and v. We fix coordinates (y1 , y2 , y3 , y4 ) in R4so that v(xk1 y k2 z k3 tk4 ) = (k1 , k2 , k3 , k4 ). The valuation v is natural from ageometric viewpoint: if v(f ) = (k1 , k2 , k3 , k4 ) then k1 is the order of vanishingof f along the hypersurface {x = 0}, while k2 is the order of vanishing of(x−k1 f )|{x=0} along the hypersurface {x = y = 0} ⊂ {x = 0} and so on.Proposition 4.1.

Define a unimodular linear transformation of R4 by theformulaϕ : (y1 , y2 , y3 , y4 ) → (−y1 , −y1 − y2 , y3 + 2y4 , y4 ).Then SP λ = ϕ(Δv (X, Lλ ))+(λ1 , λ1 +λ2 , 0, 0). In particular, Δv (X, Lλ ) canbe described by inequalities:0 ≤ y1 , y2 , y3 , y4 ;y1 ≤ λ1 ;2(y1 + y2 ) + y3 + 2y4 ≤ 2(λ1 + λ2 );y1 + y2 + y3 + 2y4 ≤ λ1 + 2λ2 ;y4 ≤ λ2Proof. Note that |SP λ ∩ Z4 | = dim Vλ as polynomials in λ by [K13, Theorem 3.6]. Comparing the highest degree homogeneous parts in λ1 and λ2 onboth sides and using Hilbert’s theorem we getvolume(SP λ ) =1deg pλ (Sp4 /B),4!where pλ : Sp4 /B → P(Vλ ) is the projective embedding of the flag varietycorresponding to the weight λ.

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