Диссертация (1136188), страница 52
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The reflection s acts on the Laurent polynomials Z[t, t−1 ] by s(tk ) = ts(k) . Define the operator TΠ on Z[t, t−1 ] by the1078Valentina Kiritchenkoformulaf − t · s(f ).1−tIt is not hard to see that, for every Laurent polynomial f , the function TΠ f isalso a Laurent polynomial. The operator TΠ depends on the parallelepipedΠ = Π(μ,ν). For a subset A ⊂ Π(μ, ν), we definethe Laurent polynomialχ(A) := x∈A∩Zn tσ(x) ∈ Z[t, t−1 ] where σ(x) := ni=1 xi .TΠ f =Proposition 2.5. [KST, Proposition 6.8] Let Γ be a reduced face of Π suchthat Γ contains the vertex (μ1 , . . . , μn ). Then⎛⎞⎛⎞⎜⎟TΠ χ ⎝F⎠ = χ⎝E⎠ .F ∈L(Γ)F ∈L(Γ)E∈M (F )Example 2.6. The simplest but crucial example is Γ = {(μ1 , .
. . , μn )}.Then L(Γ) = {Γ} and M (Γ) = E(Π). Hence, χ(Γ) = tσ(μ) and χ( E∈M (Γ) E)σ(ν)= i=σ(μ) ti . The above proposition reduces to the geometric progressionsum formula:σ(ν)tσ(μ) − t · tσ(ν)ti .=1−ti=σ(μ)It is not hard to deduce Proposition 2.5 from this partial case.In what follows, we sometimes denote mitosis on parallelepipeds by MΠto indicate which parallelepiped Π to consider.2.3. Mitosis on parapolytopesWe now use mitosis on parallelpipeds to define mitosis on a more generalclass of polytopes, namely, on parapolytopes.
Consider the space with thedirect sum decompositionR d = R d 1 ⊕ · · · ⊕ Rd rand choose coordinates x = (x11 , . . . , x1d1 ; · · · ; xr1 , . . . , xrdr ) with respect to thisdecomposition.Definition 5. A convex polytope P ⊂ Rd is called a parapolytope if for anyi = 1, . . . , r, and any vector c ∈ Rd the intersection of P with the parallelGeometric mitosis1079translate of Rdi by c is either empty or the parallel translate of a coordinateparallelepiped in Rdi , i.e.,P ∩ (c + Rdi ) = c + Π(μc , νc )for μc and νc that depend on c.Example 2.7. Consider the decomposition Rd = Rn−1 ⊕ Rn−2 ⊕ · · · ⊕ R(that is, r = n − 1 and d = n(n−1)). Let λ = (λ1 , .
. . , λn ) be a non-increasing2collection of real numbers. For every λ, define the Gelfand–Zetlin polytopeGZ λ by the inequalitiesλ1x11λ2x21x12...···λ3······..xn−21x2n−2x1n−1λn.xn−22xn−11where the notationabcmeans a ≥ c ≥ b. It is easy to check that GZ λ is a parapolytope.If P ⊂ Rd is a parapolytope then we can define r different mitosis operations M1 , . . . , Mr on faces of P . These operations come from mitosis onparallelepipeds Pλ ∩ (c + Rd1 ), . . . , Pλ ∩ (c + Rdr ), respectively. For a polytope Γ ⊂ Rd , denote by Γ◦ the relative interior of Γ, i.e., Γ◦ consists of allpoints of Γ that do not lie in faces of smaller dimension.Definition 6. Let i = 1, . . .
, r, and Γ a face of P . Choose c ∈ Γ◦ . Put Πc :=P ∩ (c + Rdi ) and Γc := Γ ∩ (c + Rdi ). The set Mi (Γ) consists of all facesΔ ⊂ P such that Δ◦ contains F ◦ for some F ∈ MΠc (Γc ). Here MΠc is themitosis on the parallelepiped Πc (see Definition 2).It is easy to check that Mi (Γ) does not depend on the choice of c ∈ Γ◦ .Similarly, we can define the L-class Li (Γ) if Γc is reduced.Definition 7. Let i = 1, . . . , r, and Γ a face of P . We say that Γ is Li reduced if Γc := Γ ∩ (c + Rdi ) is reduced for some c ∈ Γ◦ .1080Valentina KiritchenkoExample 2.8.
Consider Example 2.7 for n = 3. There will be two mitosisoperations M1 , M2 . Let us apply compositions of M1 and M2 to the vertexaλ = {x11 = λ2 ; x21 = x12 = λ3 } (i.e., the vertex with the lowest sum of coordinates). The resulting faces will all contain aλ , and hence, can be encodedby the following table:+ ⇔ x11 = λ2+ ⇔ x12 = λ3+ ⇔ x21 = x12e.g. the face {x11 = λ2 } is encoded by+.Applying Definition 6 repeatedly, we getaλ =++ M1−→+M2aλ −→+++ M2−→+M1−→++M1−→= GZ λ,+M2−→GZ λFrom a combinatorial viewpoint, this is exactly mitosis on pipe dreams of[KnM] (after reflecting our diagrams in a vertical line). For arbitrary n,geometric mitosis on GZ λ also yields combinatorial mitosis on pipe dreams(see [KST, Section 6.3]).We now consider an example where geometric mitosis produces a newcombinatorial rule.Example 2.9. Let λ = (λ1 , λ2 ), where λ1 and λ2 are positive real numbers.In [K13, Example 3.4], convex-geometric divided difference operators wereused to construct the following symplectic DDO polytope SP λ in R4 :0 ≤ y1 ≤ λ 1 ,y2 ≤ y1 + λ2 ,y3 ≤ y2 + λ2 ,0 ≤ y4 ≤ λ2 ,y3 ≤ 2y2 ,y3y4 ≤ .2As can be readily seen from the inequalities, it is a parapolytope with respect to the decomposition R4 = R2 ⊕ R2 given by x11 = y1 , x21 = y2 , x12 = y3 ,x22 = y4 .
Hence, there are two mitosis operations M1 and M2 . Again, let usapply compositions of M1 and M2 to the lowest (with respect to the sum ofGeometric mitosis1081coordinates) vertex 0 ∈ SP λ . Label faces of SP λ by diagrams as in Example 1.1. By Definition 6 we get0= ++M1+ −→+M2+0 −→M2−→+M2+ −→++M1+ −→⎧+⎨M1+ −→⎩+⎫+ ⎬+,+⎧⎨⎩⎭+⎫⎬+ ,,M2−→++⎭= SP λM2−→M1−→SP λThe combinatorics of the last example can be extended to the decom2position Rr = Rr ⊕ R2r−2 ⊕ R2r−4 ⊕ · · · ⊕ R2 (see Section 5).3. Geometric mitosis and Demazure operatorsIn this section, we discuss the relation between geometric mitosis, Demazureoperators and Schubert calculus.
We introduce a special class of parapolytopes associated with reductive groups. In particular, Gelfand–Zetlin polytopes and, more generally, polytopes constructed in [K13, Section 3] viaconvex-geometric divided difference operators belong to this class.Let G be a connected reductive group of semisimple rank r. Let α1 , . . . , αrdenote simple roots of G, and s1 , . .
. , sr the corresponding simple reflections.Fix a reduced decomposition w0 = si1 si2 · · · sid of the longest element w0 ofthe Weyl group of G. Let di be the number of sij in this decomposition suchthat ij = i. Consider the spaceRd = R d 1 ⊕ · · · ⊕ R d r .As before, we choose coordinates x = (x11 , . . . , x1d1 ; · · · ; xr1 , .
. . , xrdr ) with respect to this decomposition. We will also use an alternative labeling of coordinates (y1 , . . . , yd ) whereyd−j+1 = xipjjfor pj := {k ≥ j | sik = sij }.1082Valentina KiritchenkoExample 3.1. (a) Let G = GLn and w0 = (s1 )(s2 s1 )(s3 s2 s1 ) · · · (sn−1 · · · s1 ).and Rd = Rn−1 ⊕ Rn−2 ⊕ · · · ⊕ R. The labelingsThen r = n − 1, d = n(n−1)2of coordinates are related as follows:(y1 , y2 , . . . , yd ) = (x11 , x21 , . . . , xn−1; x12 , x22 , . . . , xn−2; · · · ; xn−1).121(b) Let G = Sp4 and w0 = s2 s1 s2 s1 (the symplectic DDO polytope SP λwas constructed in [K13, Example 3.4] using this decomposition). Then r =2, d = 4, R4 = R2 ⊕ R2 , and(y1 , y2 , y3 , y4 ) = (x11 , x21 , x12 , x22 )exactly as in Example 2.9.i ixj .
Let ΛG denote the weight lattice of G. Define thePut σi (x) = dj=1projection p of Rd to ΛG ⊗ R by the formula p(x) = σ1 (x)α1 + · · · + σr (x)αr .In what follows, we always assume that P lies in the positive octant andcontains the origin, that is, the origin is the vertex of P with the minimalsum of coordinates. Let λ be a dominant weight of G. In what follows, wed i ⊕ · · · ⊕ Rd r .identify Rd /Rdi with Rd1 ⊕ · · · ⊕ RDefinition 8.
Let i ∈ {1, . . . , r}. A parapolytope P ⊂ Rd is called (λ, i)balanced if for any c ∈ Rd /Rdi we haveσi (μc ) + σi (νc ) = (−w0 λ − p(c), αi ),where (·, αi ) is a coroot, i.e., is defined by the identity si (χ) = χ − (χ, αi )αifor all χ in the weight lattice.Example 3.2. We continue Example 3.1.(a) Let aλ := (λ1 , . . . , λn−1 ; λ1 , . . . , λn−2 ; · · · ; λ1 ) be the lowest vertexof the Gelfand–Zetlin polytope GZ λ (see Example 2.7). Let ω1 , .
. . , ωn−1denote the fundamental weights of SLn . It is easy to check that the paralleltranslate GZ λ − aλ of the Gelfand–Zetlin polytope is (λ, i)-balanced for alli ∈ {1, . . . , n − 1} and λ = (λ2 − λ1 )ω1 + · · · + (λn − λn−1 )ωn−1 .(b) Let λ be a strictly dominant weight of Sp4 . Let α1 denote the shorterroot, and α2 the longer one. Put λi = (λ, αi ) for i = 1, 2. It is easy to checkthat the symplectic DDO polytope SP λ from Example 2.9 is (λ, i)-balancedfor i = 1, 2.Definition 9. A parapolytope P ⊂ Rd is called λ-balanced if it is (λ, i)balanced for any i ∈ {1, .
. . , r}Geometric mitosis1083In particular, the polytopes considered in Examples 3.2 are λ-balanced.For certain w0 , one can construct λ-balanced polytopes using an elementaryconvex-geometric algorithm that mimics divided difference operators (see[K13, Theorem 3.6] for more details), e.g. Gelfand–Zetlin polytopes and thesymplectic DDO polytope SP λ can be constructed this way. Another sourceof λ-balanced polytopes might be provided by Newton–Okounkov polytopesof flag varieties for certain valuations. For instance, SP λ can also be realizedas the Newton–Okounkov polytope of the flag variety of Sp4 for a geometricvaluation associated with w0 (see Section 4).Remark 3.3.
The symplectic DDO polytope SP λ has 11 vertices, hence,it is not combinatorially equivalent to string polytopes for Sp4 and w0 =s1 s2 s1 s2 or s2 s1 s2 s1 defined in [L] (the latter have 12 vertices).If Pλ is a λ-balanced parapolytope, then geometric mitosis on Pλ is compatible with the action of Demazure operators Dα1 , . . . , Dαr on the groupalgebra Z[ΛG ]. Let α be a root of G. Recall that Dα acts on Z[ΛG ] as follows:D α eμ =eμ − eα esi (μ).1 − eαFor a subset A ⊂ Pλ , denote by Ac the intersection A ∩ (c + Rdi ). Let πi :di ⊕ · · · ⊕ Rdr be the projection that forgets coordinatesRd → R d 1 ⊕ · · · ⊕ Rii(x1 , .
. . , xdi ).Theorem 3.4. Let i ∈ {1, . . . , r}, and S a collection of Li -reduced faces ofa λ-balanced parapolytope Pλ that satisfy the following conditions.(1) Every F ∈ S contains the vertex 0 ∈ Pλ .(2) If F ∈ S, then Li (F ) ⊂ S.(3) For every F ∈ S with empty Mi (F ) there exists F ∈ S with nonempty and some c ∈ F ◦ .Mi (F ) such that Fc ⊂ Γc for some Γ ∈ Mi (F ) (4) The sets S := F ∈S F and Mi (S) := F ∈S E∈Mi (F ) E have thesame image under πi , i.e., πi (S) = πi (Mi (S)).Then we haveD αiew0 λx∈S∩Zdep(x)= e w0 λMi (S)∩Zdep(x) .1084Valentina KiritchenkoProof. Every x ∈ Pλ can be written uniquely as πi (x) + z where z ∈ Πc .Since p(x) = p(πi (x)) + σi (z)αi we getep(x) =x∈S∩Zdep(c)tσi (z) ,z∈Sc ∩Zdic∈πi (S)∩Zd−diwhere t := eαi .
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