Диссертация (1136188), страница 56
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161–184Разрешение на копирование: Согласно Соглашению о копирайте автор статьиможет использовать копии в образовательных и научных целях. Прикопировании требуется указать источник.71-06.tex : 2016/10/27 (14:27)page: 161Advanced Studies in Pure Mathematics 71, 2016Schubert Calculus — Osaka 2012pp. 161–184Divided difference operators on polytopesValentina KiritchenkoAbstract.We define convex-geometric counterparts of divided difference(or Demazure) operators from the Schubert calculus and representation theory. These operators are used to construct inductively polytopes that capture Demazure characters of representations of reductivegroups.
In particular, Gelfand–Zetlin polytopes and twisted cubes ofGrossberg–Karshon are obtained in a uniform way.§1.IntroductionPolytopes play a prominent role in representation theory and algebraic geometry. In algebraic geometry, there are Okounkov convexbodies introduced by Kaveh–Khovanskii and Lazarsfeld–Mustata (see[KKh] for the references). These convex bodies turn out to be polytopesin many important cases (e.g. for spherical varieties). In representationtheory, there are string polytopes introduced by Berenstein–Zelevinskyand Littelmann [BZ, L]. String polytopes are associated with the irreducible representations of a reductive group G, namely, the integerpoints inside and at the boundary of a string polytope parameterize acanonical basis in the corresponding representation.
A classical exampleof a string polytope for G = GLn is a Gelfand–Zetlin polytope.There is a close relationship between string polytopes and Okounkovbodies. String polytopes were identified with Okounkov polytopes of flagvarieties for a geometric valuation [K] and were also used in [KKh] to giveReceived July 31, 2013.Revised February 6, 2014.Key words and phrases. Gelfand–Zetlin polytope, divided difference operator, Demazure character.The author was supported by Dynasty foundation, AG Laboratory NRUHSE, MESRF grants ag. 11.G34.31.0023, MK-983.2013.1 and by RFBR grants12-01-31429-mol-a, 12-01-33101-mol-a-ved.
This study was carried out within“The National Research University Higher School of Economics’ Academic FundProgram in 2013-2014, research grant No. 12-01-0194”.71-06.tex : 2016/10/27 (14:27)162page: 162V. Kiritchenkoa more explicit description of Okounkov bodies associated with actionsof G on algebraic varieties. Natural generalizations of string polytopesare Okounkov polytopes of Bott–Samelson resolutions of Schubert varieties for various geometric valuations (an example of such a polytope iscomputed in [Anderson]).In this paper, we introduce an elementary convex-geometric construction that yields polytopes with the same properties as string polytopes and Okounkov polytopes of Bott–Samelson resolutions.
Namely,exponential sums over the integer points inside these polytopes coincide with Demazure characters. We start from a single point and applya sequence of simple convex-geometric operators that mimic the wellknown divided difference or Demazure operators from the Schubert calculus and representation theory. Convex-geometric Demazure operatorsact on convex polytopes and take a polytope to a polytope of dimension one greater. In particular, classical Gelfand–Zetlin polytopes canbe obtained in this way (see Section 3.2).
More generally, these operators act on convex chains. The latter were defined and studied in[PKh] and used in [PKh2] to prove a convex-geometric variant of theRiemann–Roch theorem.When G = GLn , convex-geometric Demazure operators were implicitly used in [KST] to calculate Demazure characters of Schubert varietiesin terms of the exponential sums over unions of faces of Gelfand–Zetlinpolytopes and to represent Schubert cycles by unions of faces. A motivation for the present paper is to create a general framework for extendingresults of [KST] on Schubert calculus from type A to arbitrary reductivegroups. In particular, convex-geometric divided difference operators allow one to use in all types a geometric version of mitosis (mitosis onparallelepipeds) developed in [KST, Section 6]. This might help to findan analog of mitosis of [KnM] in other types.Another motivation is to give a tool for describing inductively Okounkov polytopes of Bott–Samelson resolutions.
We describe polytopesthat conjecturally coincide with Okounkov polytopes of Bott–Samelsonresolutions for a natural choice of a geometric valuation (see Conjecture 4.1). Another application is an inductive description of Newton–Okounkov polytopes for line bundles on Bott towers (in particular, ontoric degenerations of Bott–Samelson resolutions) that were first described by Grossberg and Karshon [GK] (see Section 4.1 and Remark4.6).This paper is organized as follows. In Section 2, we give backgroundon convex chains and define convex-geometric divided difference operators.
In Section 3, we relate these operators with Demazure characters71-06.tex : 2016/10/27 (14:27)page: 163Divided difference operators on polytopes163and their generalizations. In Section 4, we outline possible applicationsto Okounkov polytopes of Bott towers and Bott–Samelson varieties.I am grateful to Dave Anderson, Joel Kamnitzer, Kiumars Kavehand Askold Khovanskii for useful discussions. I would also like to thankthe referee and Megumi Harada for the careful reading and valuablecomments.§2.Main construction2.1.
String spaces and parapolytopesDefinition 1. A string space of rank r is a real vector space Rdtogether with a direct sum decompositionRd = Rd1 ⊕ . . . ⊕ Rdrand a collection of linear functions l1 , . . . , lr ∈ (Rd )∗ such that li vanisheson Rdi .We choose coordinates in Rd such that they are compatible withthe direct sum decomposition. The coordinates will be denoted by(x11 , . .
. , x1d1 ; . . . ; xr1 , . . . , xrdr ) so that the summand Rdi is given by vanishing of all coordinates except for xi1 ,. . . , xidi . In what follows, we regardRd as an affine space.Let μ = (μ1 , . . . , μdi ) and ν = (ν1 , . . . , νdi ) be two collections ofreal numbers such that μj ≤ νj for all j = 1,. . . , di . By the coordinateparallelepiped Π(μ, ν) ⊂ Rdi we mean the parallelepipedΠ(μ, ν) = {(xi1 , . .
. , xidi ) ∈ Rdi | μj ≤ xij ≤ νj , j = 1, . . . , di }.Definition 2. A convex polytope P ⊂ Rd is called a parapolytope iffor i = 1,. . . , r, and any vector c ∈ Rd the intersection of P with theparallel translate c + Rdi of Rdi is either empty or the parallel translateof a coordinate parallelepiped, i.e.,P ∩ (c + Rdi ) = c + Π(μc , νc )for μc and νc that depend on c.For instance, if d = r (i.e., d1 = . . . = dr = 1) then every polytopeis a parapolytope. Below is a less trivial example of a parapolytope ina string space.Example 2.1. Consider the string spaceRd = Rn−1 ⊕ Rn−2 ⊕ .
. . ⊕ R171-06.tex : 2016/10/27 (14:27)page: 164164V. Kiritchenkoof rank r = (n − 1) and dimension d = n(n−1).2Let λ = (λ1 , . . . , λn ) be a non-increasing collection of integers. Foreach λ, define the Gelfand–Zetlin polytope Qλ by the inequalitiesλ1λ2x11x21λ3...x12x2n−2.....λn......xn−21x1n−1.xn−22xn−11where the notationabcmeans a ≥ c ≥ b. It is easy to check that Qλ is a parapolytope. Indeed, consider the parallel translate of Rn−i by a vector). Put c0i = λi for i = 1,. . . , n.
The inc = (c11 , . . . , c1n−1 ; . . . ; cn−11n−iis given by the the following inequalities:tersection of Qλ with c + Rci−11xi1ci−12ci+11xi2ci−13.........ci+1n−i−1xin−ici−1n−i+1.Therefore, the intersection can be identified with the coordinate pari+1allelepiped c + Π(μ, ν) ⊂ c + Rn−i , where μj = max(ci−1j , cj−1 ) andi+1i+1νj = min(ci−1= −∞ and ci+1j+1 , cj ) (put c0n−i = +∞).2.2. Polytopes and convex chainsConsider the set of all convex polytopes in Rd . This set can be endowed with the structure of a commutative semigroup using MinkowskisumP1 + P2 = {x1 + x2 ∈ Rd | x1 ∈ P1 , x2 ∈ P2 }It is not hard to check that this semigroup has cancelation property. Wecan also multiply polytopes by positive real numbers using dilation:λP = {λx | x ∈ P },λ ≥ 0.Hence, we can embed the semigroup S of convex polytopes into itsGrothendieck group V , which is a real (infinite-dimensional) vectorspace.
The elements of V are called virtual polytopes.It is easy to check that the set of parapolytopes in Rd is closed underMinkowski sum and under dilations. Hence, we can define the subspaceV ⊂ V of virtual parapolytopes in the string space Rd .71-06.tex : 2016/10/27 (14:27)page: 165Divided difference operators on polytopes165Example 2.2. If Rd is a string space of rank 1, i.e.
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