Диссертация (1136188), страница 51

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The combinatorics of mitosis depends significantly onthe combinatorics of Pλ . For instance, for Gelfand–Zetlin polytopes we getmitosis on usual pipe dreams, and for a polytope associated with the coneof adapted strings in type C (see Section 5) we get different combinatorialobjects that we call skew pipe dreams.Example 1.1.

Let SP λ ⊂ R4 be the Newton–Okounkov polytope of theflag variety X = Sp4 /B and the line bundle Lλ for the valuation associated with the flag s2 s1 s2 s1 Xid ⊂ s2 s1 s2 Xs1 ⊂ s2 s1 Xs2 s1 ⊂ s2 Xs1 s2 s1 ⊂ X oftranslated Schubert subvarieties (see Section 4.1 for more details on Newton–Okounkov polytopes). Here s1 and s2 denote the reflections associated withthe shorter and longer simple roots, respectively. One can show that thispolytope is defined by 8 inequalities and moreover one can choose coordinates (y1 , y2 , y3 , y4 ) in R4 so that precisely 4 inequalities become homogeneous, namely, 0 ≤ y1 , 0 ≤ 2y4 ≤ y3 ≤ 2y2 (see Example 2.9).

However, SP λis combinatorially different from the string polytope (see Remark 3.3). Thefaces of SP λ that contain 0 can be encoded by the following diagram:+ ⇐⇒ 0 = y1+ ⇐⇒ 0 = y4+ ⇐⇒ y4 = y23+ ⇐⇒ y3 = 2y2e.g. {y1 = 0, y3 = 2y2 } is encoded by +.+The Schubert cycles on Sp4 /B can be represented by the following unionsof faces of SP λ .Sid = {0} = ++Ss1 s2 =+++ , Ss1 =++∪+, Ss2 s1 =++ , Ss2 = ++++ ,++ , Ss1 s2 s1 =+,1074Ss2 s1 s2 =Valentina Kiritchenko+∪+∪+, Ss2 s1 s2 s1 = SP λ =.Geometric mitosis provides a simple combinatorial rule to generate all thesediagrams starting from the top one (see Section 5).This paper is organized as follows. In Section 2, we recall mitosis onparallelepipeds and its relation to Demazure-type operators, and define geometric mitosis on parapolytopes.

In Section 3, we consider parapolytopes associated with reductive groups and prove Theorem 3.4 that relates Demazureoperators with geometric mitosis. In Corollary 3.6, we give an algorithm forgenerating faces that represent a given Demazure character (or equivalently,a given Schubert cycle). In Section 4, we apply the results of the precedingsections to Sp4 and the symplectic DDO polytope SP λ constructed in [K13].We prove that SP λ can be realized as the Newton–Okounkov body of theflag variety Sp4 /B and the line bundle Lλ for a natural geometric valuationconsidered in [An, Ka13]. Next, we outline how results of [Ka11, KST] can beused to model the Schubert calculus on Sp4 /B by intersecting faces of SP λ .In Section 5, we describe combinatorics of geometric mitosis, in particular,define mitosis on skew pipe dreams in type C that generalize combinatoricsof mitosis for SP λ .

We also formulate open questions.2. Mitosis on polytopesIn this section, we define a convex-geometric operation (geometric mitosis) on faces of polytopes that models Demazure operators from representation theory. The definition is elementary and reduces to the case of parallelepipeds, which we discuss first. For special classes of polytopes associated with reductive groups, geometric mitosis has algebro-geometric andrepresentation-theoretic meaning. This will be discussed in the next section.2.1. Mitosis on pipe dreamsFor motivation, we first recall briefly mitosis on pipe dreams introduced in[KnM, M] as a positive rule for listing inductively (without cancelation andredundancy) all monomials in a given Schubert polynomial. A pipe dreamis a network of pipes obtained by tiling a square grid Z>0 × Z>0 by finitely✞ (elbow joint).

Every pipemany square tiles of two types:(cross) and ✆dream defines a permutation w by the following rule: the pipe entering rowGeometric mitosis1075i exits from column w(i) (see [M, Figure 1] for an example). A pipe dreamis reduced if every pair of pipes crosses at most once (reduced pipe dreamsare also called rc-graphs).Consider pipe dreams with permutations from Sn . By replacing crosseswith pluses and erasing elbow joints, one can represent them by square n × ntables whose cells are either filled with + or empty. Let i = 1, . . . , n − 1.

Thei-th mitosis is an operation on rows i and i + 1 of pipe dreams. It assigns toevery reduced pipe dream with permutation w a collection (possibly empty)of reduced pipe dreams with permutation wsi where si = (i i + 1) (see [M,Definition 6] for more details). Let RP(w) denote the set of all reduced pipedreams with a given permutation w ∈ Sn .

Mitosis gives an efficient rule tolist all pipe dreams in RP(w) because of the following identity [M, Theorem15]:RP(wsi ) =mitosisi (D)D∈RP(w)whenever l(wsi ) = l(w) − 1.2.2. Mitosis on parallelepipedsRecall the mitosis on parallelepipeds (or paramitosis) from [KST, Section 6]using more geometric terms. Let Π := Π(μ, ν) ⊂ Rn be a parallelepiped givenby inequalities μi ≤ xi ≤ νi for i = 1, . . . , n. In what follows, we will onlyconsider parallelepipeds of this kind.

They will be called coordinate parallelpipeds.Definition 1. An edge of Π is essential if it is given by equationsx1 = μ1 , . . . , xi−1 = μi−1 ; xi+1 = νi+1 , . . . , xn = νn .Clearly, the number of essential edges is equal to dim Π, and the unionof essential edges forms a broken line that connects the vertices (μ1 , . . . , μn )and (ν1 , . .

. , νn ). Denote the set of essential edges of Π by E(Π).For every face Γ ⊂ Π, we now define a collection of faces M (Γ). Let k bethe minimal number such that Γ ⊆ {xi = μi } for all i > k (in particular, Γ {xk = μk }) and νi = μi for at least one i > k. If no such k exists then M (Γ) =∅. Under the isomorphism Rn Rk × Rn−k ; (x1 , . . . , xn ) → (x1 , . . . , xk ) ×(xk+1 , . . . , xn ) the parallelepiped Π gets mapped to Π × Π where Π ⊂ Rkand Π ⊂ Rn−k are coordinate parallelepipeds.

The face Γ gets mapped toΓ × v where v = (μk+1 , . . . , μn ) is a vertex of Π and Γ ⊂ Π is a face of Π .1076Valentina KiritchenkoDefinition 2. The set M (Γ) (called the mitosis of Γ) consists of all facesΓ × E such that E is an essential edge of Π .In particular, dim Δ = dim Γ + 1 for any Δ ∈ M (Γ). It is easy to checkthat M 2 (Γ) = ∅ for any face Γ.

Here is a key example of mitosis.Example 2.1. If Γ is the vertex (μ1 , . . . , μn ), then M (Γ) is the set ofessential edges of Π.This geometric version of mitosis is similar to the combinatorial mitosis of [KnM]. To see this represent every face of Π(μ, ν) by a 2 × n table(aij )i=1,2, 1≤j≤n whose cells are either filled with + or empty. Namely, theface satisfies the equality xi = μi or xi = νi if and only if a1i = + or a2i = +,respectively (in particular, if μi = νi then the i-th column has two +).

Onthe level of tables, operation M coincides with the i-th mitosis of [KnM] onreduced pipe dreams restricted to the rows i and i + 1 after reflecting ourtables in a vertical line (cf. [M, Definition 6]).Example 2.2. If Π(μ, ν) ⊂ R4 , where μ = (1, 1, 1, 1) and ν = (2, 2, 1, 2)(that is, μ3 = ν3 ), then the edge Γ = { x2 = μ2 , x4 = μ4 } is representedby the table++++The set M (Γ) consists of two edges represented by the following tables:+++,+++The combinatorial notion of chute moves and ladder moves on rc-graphsor pipe dreams introduced in [BB] (cf. [M, Definition 8]) can also be extendedto the geometric setting as follows.Definition 3. A face Γ ⊂ Π is called reduced if its table does not containa square or the form+.+We now describe reduced faces geometrically. For a partition J = (0 ≤j1 < · · · < jk−1 < jk ≤ n), let pJ,i denote the projection (x1 , .

. . , xn ) →Geometric mitosis1077(xji +1 , . . . , xji+1 ) for i = 1, . . . , k − 1. For i = 0 and i = k the projections pJ,iare defined by (x1 , . . . , xn ) → (x1 , . . . , xj1 ) and (x1 , . . . , xn ) → (xjk +1 , . . . , xn ),respectively. It is easy to check that Γ is reduced if and only if there is apartition J(Γ) such that pJ,1 (Γ) and pJ,k+1 (Γ) are the vertices (ν1 , . . . , νj1 −1 )and (μjk +1 , .

. . , μn ), respectively, and pJ,i (Γ) ∈ E(pJ,i (Π)) for any i = 2,. . . , k.A partition J(Γ) is unique if we ignore all indices i ∈ {1, . . . , n} suchthat μi = νi . In the language of [M], the partition J(Γ) corresponds to thedecomposition of pipe dreams into introns (cf. [M, Definition 11]). The lengthof J(Γ) is equal to dim Γ + 2.Definition 4. Two reduced faces Γ and Γ are said to be L-equivalent ifJ(Γ) = J(Γ ).

Denote by L(Γ) the set of all reduced faces equivalent to Γ.Example 2.3. Take Π(μ, ν) ⊂ R5 , where μ = (1, 1, 1, 1, 1) and ν = (2, 2, 1,2, 2). The face Γ = {x1 = μ1 ; x4 = ν4 } of dimension 2 is reduced with respectto the partition (0, 4, 5). The set L(Γ) consists of three faces:+ + ++++,(= Γ),.++ ++ + +Remark 2.4.

There is a bijection between L-equivalence classes and facesof the standard simplex (see [KST, Proposition 6.6]), which yields a minimalrealization of the simplex as a cubic complex. Using this bijection it is nothard to check that for any face Γ ⊂ Π the mitosis applied to faces in L(Γ)produces a single L–equivalence class, i.e.,E = L(Γ )F ∈L(Γ)E∈M (F )for some face Γ ⊂ Π (see [KST, Remark 6.7] for more details).Definitions of M (Γ) and L(Γ) are motivated by the identity [KST,Proposition 6.8] for a Demazure-type operator applied to an exponentialsum over Γ. We briefly recall this identity (for more details see [KST, Section 6]). Let s : Z → Z be a reflection aboutn1C :=(μi + νi ),2i=1that is, s(k) = 2C − k for k ∈ Z.

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