Диссертация (1136188), страница 55
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Note that from now on s1 corresponds to the longer root in2accordance with [L]. Let P ⊂ Rn be a parapolytope with the lowest vertex0 such that the vertex cone C0 is defined by inequalities(*)0 ≤ xi2 ≤ xi−1≤ xi−2≤ · · · ≤ x22i−2 ≤ x1i ≤ x22i−346≤ · · · ≤ xi−2≤ xi−1≤ xi153for all i = 1, . . . , n. There are n2 inequalities in (∗), in particular, 0 is asimple vertex of C0 . The cone C0 is exactly the cone of adapted strings forthe decomposition w0 (see [L, Theorem 6.1]).Geometric mitosis1093Faces of C0 can be encoded by skew pipe dreams.
A skew pipe dreamof size n is a (2n − 1) × n table whose cells are either empty or filled with+. Only cells (i, j) with n − j < i < n + j are allowed to have +. Whendrawing a skew pipe dream we omit cells (i, j) that do not satisfy theseinequalities. For instance, all tables of Example 2.9 are skew pipe dreamsof size n = 2. There is a bijective correspondence between faces of C0 andskew pipe dreams: to get the skew pipe dream D(Γ) corresponding to a faceΓ ⊂ C0 replace an inequality xkl ≤ xkl (or 0 ≤ xkl ) in (∗) by + at cell⎧if l is odd, k = 1⎨ (n + k − 1, k + l−12 )l(**)(n − k + 1, k + 2 − 1) if l is even, k = 1⎩(n, l)if k = 1whenever xkl = xkl (or 0 = xkl ) identically on Γ. Table (∗∗) gives a bijectionbetween coordinates xkl and (fillable) cells of a skew pipe dream.Example 5.1. Let n = 3.
The bijection between cells and coordinates givenby (∗∗) is depicted on the left. The skew pipe dream D(G) of the faceΓ = {0 = x11 ; 0 = x22 = x12 ; 0 = x32 ; x23 = x31 } is depicted on the right.x11x22x12x21x32x24x13x23x31+++++The bijection between faces of C0 and skew pipe dreams transformsgeometric mitosis on faces of C0 into the following combinatorial rule. Weuse terminology of [M, Section 3]. Given a skew pipe dream D of size n,definestarti (D) = min{Sn−i+1 , Sn+i−1 + 1},where Sj denotes the column index of the leftmost empty cell in row j, i.e.,/ D),starti (D) = min{ min(j | (n − i + 1, j) ∈min(j | (n + i − 1, j) ∈/ D) + 1},so the (n ± (i − 1))-th rows of D are filled solidly with crosses in the regionto the right and upward of cell (starti (D) − 1, n + i − 1). Let/ D}J − (D) = {columns j strictly to the right of starti (D) | (n − i + 2, j) ∈1094Valentina KiritchenkoandJ + (D) = {columns j strictly to the right of starti (D) | (n + i, j) ∈/ D}.For p ∈ J ± (D), we now construct the offspring Dp± in two or three steps asfollows.1) If p ∈ J − (D), to construct Dp− delete the cross at (n − i + 1, p) fromD.
If p ∈ J + (D), to construct Dp+ delete the cross at (n + i − 1, p).2) Take all crosses in row n − i + 1 of J − (D) and in row n + i − 1 ofJ + (D) that are to the right of column p, and move each one down tothe empty box below it in row n − i + 2 and in row n + i, respectively.3) If p ∈/ J − (D) ∩ J + (D) or i = 1, then we are done with both Dp− and+Dp . Otherwise, an additional step is required to construct Dp+ : movethe cross at (n − i + 1, p) to the empty box below it in row n − i + 2.Definition 11. The i-th mitosis operator sends a skew pipe dream D tomitosisi (D) = {Dp− | p ∈ J − (D)} ∪ {Dp+ | p ∈ J + (D)}.Note that the i-th mitosis affects only rows n ± (i − 1), n − i + 2 andn + i, and mitosisi (D) is empty if both J + and J − are empty.
It is easyto check that under the above bijection between faces of C0 and skew pipedreams we havemitosisi (D(Γ)) = Mi (Γ).In particular, for n = 2 this combinatorial algorithm yields exactly the sametables as in Example 2.9.Example 5.2. Let n = 3 and i = 2.+++D=+mitosis2−→⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩++mitosis2−→+++D2+=++ ,++D2−=++ ,++D3−⎫+ ⎪⎪⎪⎪⎬=+++⎪⎪⎪⎪⎭Geometric mitosis1095In this example, starti (D) = 1, J − (D) ={columns 2, 3} and J + (D) ={column 2}.5.3. Open questionsIt is tempting to use combinatorial mitosis on skew pipe dreams to produce an explicit realization of generalized Newton–Okounkov polytopes forSchubert varieties on Sp2n /B by collections of faces of symplectic stringpolytopes. While such a realization exists by general properties of stringpolytopes (see [Mi, Section 5.5] for more details) an explicit description isknown only for n = 2 (see [I]). However, the symplectic string polytopes associated with w0 = (sn sn−1 · · · s2 s1 s2 · · · sn−1 sn ) · · · (s2 s1 s2 )(s1 ) are not para2polytopes with respect to the decomposition Rn = Rn ⊕ R2n−2 ⊕ R2n−4 ⊕· · · ⊕ R2 (already for n = 2), so Corollary 3.6 can not be directly applied tothem.As we have seen in Section 4, the symplectic DDO polytope in the case ofSp4 turned out to be a more suitable candidate for constructing explicit generalized Newton–Okounkov polytopes using Corollary 3.6.
Symplectic DDOpolytopes can also be constructed for Sp2n using reduced decompositionw0 = (sn · · · s1 )n rather than w0 (note that for n = 2 we have w0 = w0 ).In an ongoing project with M. Padalko, we aim to describe these polytopesexplicitly by inequalities, study combinatorics of their geometric mitosis andapplications to the Schubert calculus on Sp2n /B.It is also interesting to check whether the Newton–Okounkov polytopesof flag varieties associated with the lowest term valuation v w0 (see Section 4)are good candidates for applying geometric mitosis to the Schubert calculus.Proposition 4.1 suggests that this might be the case.
Recall that theory ofNewton–Okounkov polytopes can be used to construct toric degenerations[An]. If a Newton–Okounkov polytope P of the flag variety X satisfies conditions of Corollary 3.6 and XP is the toric degeneration of X associatedwith P then it is natural to expect that collections of faces given by geometric mitosis yield degenerations of Schubert varieties to (reduced) toricsubvarieties of XP .AcknowledgementsI am grateful to Dave Anderson, Megumi Harada and Kiumars Kaveh foruseful discussions.
I would also like to thank the referee for valuable suggestions. I was partially supported by a subsidy granted to the HSE by the1096Valentina KiritchenkoGovernment of the Russian Federation for the implementation of the GlobalCompetitiveness Program.References[A] H. H. Andersen, Schubert varieties and Demazure’s character formula, Invent. Math. 79 (1985), no.
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Kogan and E. Miller, Toric degeneration of Schubert varietiesand Gelfand-Tsetlin polytopes, Adv. Math. 193 (2005), no. 1, 1–17.[KnM] A. Knutson and E. Miller, Gröbner geometry of Schubert polynomials, Ann. of Math. (2) 161 (2005), 1245–1318.[L] P. Littelmann, Cones, crystals and patterns, Transform. Groups 3(1998), pp. 145–179.[M] E. Miller, Mitosis recursion for coefficients of Schubert polynomials,J. Comb. Theory A 103 (2003), no. 2, 223–235.[Mi] J. Miller, Okounkov bodies of Borel orbit closures in wonderful groupcompactifications, PhD Thesis, Ohio State University, 2014.Laboratory of Algebraic Geometry and Faculty of MathematicsNational Research University Higher School of EconomicsUsacheva Str. 6, 119048, Moscow, Russia& Institute for Information Transmission Problems, Moscow, RussiaE-mail address: vkiritch@hse.ruReceived May 20, 2015Приложение I.Статья 9.Valentina Kiritchenko “Divided difference operators on polytopes”Advanced Studies in Pure Mathematics 71, 2016 Schubert Calculus — Osaka2012 pp.
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