Диссертация (1136188), страница 54

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

Текст из файла (страница 54)

Hence, to compare SP λ and Δ :=ϕ(Δv (X, Lλ )) + (λ1 , λ1 + λ2 , 0, 0) it is enough to show that SP λ ⊂ Δ. Sinceboth convex bodies have the same volumes the inclusion will imply the exactequality.We now check that SP λ ⊂ Δ. There is a natural embedding X → P3 ×IG(2, 4); (a, l) ∈ a×l, where IG(2, 4) is the Grassmannian of isotropic planesin C4 . Let pω1 , pω2 denote the projections of X to the first and secondfactor, respectively.

Then Lω1 = p∗ω1 OP3 (1) and Lω2 = p∗ω2 π ∗ OP4 (1) where π :IG(2, 4) → P4 is the Plücker embedding. Hence, H 0 (X, Lω1 ) = 1, −x, y +xz, z and H 0 (X, Lω2 ) = 1, −(y + 2xz + x2 t), z + xt, yt − z 2 , t. By takingGeometric mitosis1089the lowest order terms of basis sections we get that Δv (X, Lω1 ) contains thesimplex with the vertices(0, 0, 0, 0), (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0)and Δv (X, Lω2 ) contains the simplex with the vertices(0, 0, 0, 0), (0, 1, 0, 0), (0, 0, 2, 0), (0, 0, 0, 1).It is easy to check that ϕ takes these two simplices to SP ω1 − (1, 1, 0, 0) and⊗λ21SP ω2 − (0, 1, 0, 0), respectively. Since Lλ = L⊗λω1 ⊗ Lω2 the super-additivityof Newton–Okounkov bodies (see [KaKh, Theorem 4.9(3)]) implies thatΔv (X, Lλ ) contains the Minkowski sum λ1 Δv (X, Lω1 ) + λ2 Δv (X, Lω2 ).�Hence, Δ contains λ1 SP ω1 + λ2 SP ω2 = SP λ as desired.Example 4.2. Take λ = ρ, i.e., λ1 = λ2 = 1.

The projective embeddingpρ : Sp4 /B → P(Vρ ) comes from the composition of mapsid×πSegreSp4 /B → P3 × IG(2, 4) −→ P3 × P4 −→ P19 .The image of Sp4 /B is contained in P(Vρ ) ⊂ P19 . In coordinates (x, y, z, t),the embedding Sp4 /B → P(Vρ ) ⊂ P19 takes the point (x, y, z, t) to⎛⎞1⎜ −x ⎟ ⎜⎟⎝y + xz ⎠ × 1z−(y + 2xz + x2 t)z + xtyt − z 2tApplying the valuation v we get all 16 = dim Vρ integer points in SP ρ (vertices of SP ρ are underlined).(0, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 2, 0), (0, 0, 0, 1),(1, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 0), (1, 0, 2, 0), (1, 0, 0, 1),(0, 2, 0, 0), (0, 1, 1, 0), (0, 1, 2, 0), (0, 1, 0, 1),(0, 0, 3, 0), (0, 0, 1, 1).Remark 4.3. Similar arguments can be applied to the valuation v corresponding to the decomposition w0 = s2 s1 s2 s1 , i.e., to the flag of translated1090Valentina KiritchenkoSchubert subvarietiess2 s1 s2 s1 Xid ⊂ s2 s1 s2 Xs1 ⊂ s2 s1 Xs2 s1 ⊂ s2 Xs1 s2 s1 ⊂ X.It is easy to check that the Newton–Okounkov body Δv (X, Lλ ) is obtainedfrom Δv (X, Lλ ) by the unimodular linear transformation (y1 , y2 , y3 , y4 ) →(y4 , y3 , y2 , y1 ).

This agrees with the fact that symplectic DDO polytopescorresponding to w0 and w0 are also the same up to an affine transformation(see [K13, Example 3.4]).Remark 4.4. Let vw0 be the highest term valuation associated with the flagof Schubert subvarieties corresponding to the terminal subwords of w0 . In[Ka13], the Newton–Okounkov bodies of flag varieties for vw0 were identifiedwith string polytopes of [L].

For Sp4 and w0 = s1 s2 s1 s2 , this is the flag Xid ⊂Xs2 ⊂ Xs1 s2 ⊂ Xs2 s1 s2 ⊂ X. By Remark 3.3, the polytopes Δvw0 (X, Lλ ) andΔv (X, Lλ ) are not combinatorially equivalent (they have different numberof vertices). In particular, one can not expect a straightforward relationbetween valuations vw0 and v w0 := v (cf. [Ka13, Remark 2.3]).4.3. Newton–Okounkov polytopes of Schubert varietiesWe now identify (the unions of) faces of SP λ obtained in Example 2.9 withgeneralized Newton–Okounkov polytopes of Schubert subvarieties of X.

Thisallows us to extend results of [KST] on Schubert calculus in terms of polytope rings from Gelfand–Zetlin polytopes and GLn to the symplectic DDOpolytope SP λ and Sp4 . A different extension was previously obtained in[I] for the string polytopes of Sp4 associated with w0 = s2 s1 s2 s1 (this polytope coincides up to a unimodular change of coordinates with the symplecticGelfand–Zetlin polytope [L, Corollary 6.2]).We say that the union of faces Δw = F ⊂SP λ F is a generalized Newton–Okounkov polytope of a Schubert subvariety Xw if |Δw ∩ Z4 | = dim H 0 (Xw ,Lλ |Xw ) as polynomials in λ.

In particular, SP λ = Δw0 and any vertex ofSP λ is a valid choice for Δid . Corollary 3.6 immediately yields the followingchoices for the other Schubert varieties:Δs1 = H2+ ∩ H3+ ∩ H4 ;Δs2 = H1+ ∩ H3+ ∩ H4+ ;Δs2 s1 s2 = H1+ ∪ H2+ ∪ H3+ ,Δs2 s1 = H3+ ∩ H4+ ;Δs1 s2 s1 = H4+ ;Δs1 s2 = (H1+ ∩ H4 )+ ∪ (H2+ ∩ H4+ );where H1+ , .

. . , H4+ denote the facets of SP λ given by equations y1 = 0, 2y2 =y3 , y3 = 2y4 , y4 = 0, respectively. Applying results of [KST, Section 2] andGeometric mitosis1091[Ka11, Theorem 4.1] to SP λ we can multiply Schubert cycles in H ∗ (X, Z)by intersecting their generalized Newton–Okounkov polytopes if the latterare transverse. For instance,[Xs1 s2 s1 ] · [Xs2 s1 s2 ] = [Δs1 s2 s1 ∩ Δs2 s1 s2 ] = [Δs1 s2 ∪ Δs2 s1 ] = [Xs1 s2 ] + [Xs2 s1 ].Using techniques of [KST, Section 2] we can realize the Schubert calculus onX in terms of SP λ . Namely, [KST, Formula (1)] gives four linear relationsbetween (equivalence classes of) facets of SP λ :[H1+ ] + [H2− ] = [H1− ];2[H2+ ] + [H3− ] = [H2− ];[H2+ ] + [H3− ] = [H3+ ];2[H3+ ] + [H4− ] = [H4+ ],where H1+ , . .

. , H4+ denote the facets of SP λ given by equations y1 = λ1 ,y2 = y1 + λ2 , y3 = y2 + λ2 , y4 = λ2 , respectively. Using these relations wecan get new generalized Newton–Okounkov polytopes, e.g.Δs1 s2 s1 = H2− ∪ H3− ∪ H4− ;Δs2 s1 s2 = H1− ,such that the intersections Δv ∩ Δw are transverse for all v and w.5. Combinatorics of geometric mitosis and open questionsWe now discuss combinatorics of mitosis on admissible balanced parapolytopes. We outline a combinatorial algorithm for generating faces that appearin Corollary 3.6. For Gelfand–Zetlin polytopes, this algorithm reduces to themitosis of [KnM] on pipe dreams.

Generalizing combinatorics of Example 2.9we define mitosis on skew pipe dreams for Sp2n . In the end of this section,we formulate open questions.5.1. Mitosis on vertex coneLet P ⊂ Rd be an admissible λ-balanced parapolytope with the lowest vertex0. Since the faces that appear in Corollary 3.6 are obtained from the vertex0 ∈ P by mitosis operations they contain 0. Hence, to describe these facesit is enough to consider the combinatorics of the vertex cone C0 of P at 0and not the whole P . Recall that the vertex cone Ca of a vertex a ∈ P bydefinition consists of all b ∈ Rd such that a + λ(b − a) ∈ P for some λ ≥ 0.Let H1 , .

. . , Hd be the facets of C0 . Note that d ≥ d, and 0 is a simple vertexof P if and only if d = d . Facets Hj correspond to homogeneous inequalitieslj ≥ 0 that define C0 .1092Valentina KiritchenkoiFix i ∈ {1, . . . , r} and consider c ∈ Rd /Rd . Since P is a parapolytopewe have that Πc := P ∩ (c + Rdi ) is given by inequalities μij (c) ≤ xij ≤ νji (c)for j = 1, . . . , id , where μij (c) are linear functions. If P is admissible thenthe parallelepiped Π0 is a segment (or a point if (λ, α1 ) = 0) given by inequality 0 ≤ xi1 ≤ (λ, α1 ) and equalities xij = 0 for j = 2, .

. . , di . So μi1 (0) =μij (0) = νji (0) = 0 for all j ≥ 2, and functions μij (c) and νji (c) are all homogeneous except for possibly ν1i (c). In particular, Uc := C0 ∩ (c + Rdi ) is givenby inequalities μij (c) ≤ xij ≤ νji (c) for j = 2, . . . , id and μi1 ≤ xij , that is, Ucis almost a parallelepiped: it might be not bounded only in xi1 -direction(if ν1i (c) = μi1 (c)). Note that mitosis on parallelepipeds defined in Section 2never produces faces that lie in the facet x1 = ν1 (unless μ1 = ν1 ). Hence,the definition of mitosis on parallelepipeds goes verbatim for the faces of Uc .Let Γ ⊂ C0 be a face. Choose c ∈ Γ◦ .

Choose facets Hj1 , . . . , Hj of C0such that every face of Uc can be uniquely represented as the intersectionof Uc with some of these facets. In particular, Γc = Hi1 ∩ · · · ∩ Hik ∩ Uc forsome {i1 , . . . , ik } ⊂ {j1 , . . . , j }, hence, Γ = Hi1 ∩ · · · ∩ Hik ∩ P . Then mitosis on Uc tells us which facets in Γ = Hi1 ∩ · · · ∩ Hik should be deleted andwhich facets added in order to get all faces in Mi (Γ). We get a purely combinatorial operation Mi on the subsets of the set {Hi1 , .

. . , Hi }. Facets of C0and all operations Mi can be encoded by diagrams similar to pipe dreams.Usual pipe dreams correspond to the case when C0 is a vertex cone of theGelfand–Zetlin polytope, or equivalently, C0 is the cone of adapted stringsin type A (see [L, Theorem 5.1]).Below we consider a new combinatorial algorithm that arises from thegeometric mitosis on the cone of adapted strings in type C.5.2. Mitosis on skew pipe dreamsLet G = Sp2n , i.e., r = n and d = n2 . Take the reduced decomposition w0 =2(sn sn−1 · · · s2 s1 s2 · · · sn−1 sn ) · · · (s2 s1 s2 )(s1 ). Then Rn = Rn ⊕ R2n−2 ⊕R2n−4 ⊕ · · · ⊕ R2 .

Характеристики

Список файлов диссертации