Диссертация (1136188), страница 57

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d1 = d, thenparapolytopes are coordinate parallelepipeds Π(μ, ν). Clearly,Π(μ, ν) + Π(μ , ν ) = Π(μ + μ , ν + ν ).Hence, virtual parapolytopes can be identified with the pairs of vectorsμ, ν ∈ Rd . This yields an isomorphism V Rd ⊕ Rd . Under thisisomorphism, the semigroup of (true) coordinate parallelepipeds getsmapped to the convex cone in Rd ⊕ Rd given by the inequalities μi ≤ νifor i = 1,.

. . , d.We now define the space Ṽ of convex chains following [PKh]. Aconvex chain is a function on Rd that can be represented as a finitelinear combinationc P IP ,Pwhere cP ∈ R, and IP is the characteristic function of a convex polytopeP ⊂ Rd , that is,1, x ∈ PIP (x) =.0, x ∈/PThe semigroup S of convex polytopes can be naturally embedded intoṼ :ι : S → Ṽ ; ι : P → IPIn what follows, we will work in the space of convex chains and freelyidentify a polytope P with the corresponding convex chain IP . However,note that the embedding ι is not a homomorphism, that is, IP +Q = IP +IQ (the sum of convex chains is defined as the usual sum of functions).Remark 2.3.

The embedding ι : S → Ṽ can be extended to thespace V of all virtual polytopes. Namely, there exists a commutativeoperation ∗ on Ṽ (called product of convex chains) such thatIP +Q = IP ∗ IQ(M )for any two convex polytopes P and Q (see [PKh, Section 2, PropositionDefinition 3]). Virtual polytopes can be identified with the convex chainsthat are invertible with respect to ∗.Similarly to the space of convex chains, define the subspace Ṽ ⊂ Ṽof convex parachains using only parapolytopes instead of all polytopes.We will use repeatedly the following example of a parachain.71-06.tex : 2016/10/27 (14:27)page: 166166V. KiritchenkoExample 2.4. Consider the simplest case d = 1.

Let [μ, ν] ⊂ Rbe a segment (i.e., μ < ν), and [ν, μ] — a virtual segment. Using theexistence of the operation ∗ satisfying (M ), it is easy to check thatι([ν, μ]) = −I[−ν,−μ] + I{−ν} + I{−μ}(note that the right hand side is the characteristic function of the openinterval (−ν, −μ)). Indeed,I[μ,ν] ∗ −I[−ν,−μ] + I{−ν} + I{−μ}= −I[μ,ν] ∗ I[−ν,−μ] + I[μ,ν] ∗ I{−ν} + I[μ,ν] ∗ I{−μ}= −I[μ−ν,ν−μ] + I[μ−ν,0] + I[0,ν−μ] = I{0} .More generally, if P ⊂ Rd is a convex polytope then(−1)dim P IP ∗ Iint(P ∨ ) = I{0} ,where P ∨ = {−x | x ∈ P }, and int(P ∨ ) denotes the interior of P ∨ (see[PKh, Section 2, Theorem 2]).2.3.

Divided difference operators on parachainsFor each i = 1,. . . , r, we now define a divided difference (or Demazure) operator Di on the space of convex parachains Ṽ . Let P be aparapolytope. Choose the smallest j = 1,. . . , di such that P lies in thehyperplane {xij = const}. If no such j exists, then Di (IP ) is not defined.Otherwise, we expand P in the direction of xij as follows.First, suppose that a parapolytope P lies in (c+Rdi ) for some c ∈ Rd ,i.e., P = c + Π(μ, ν) is a coordinate parallelepiped.

We always fix thechoice of c by requiring that c lies in the direct complement to Rdi withrespect to the decomposition Rd = Rd1 ⊕ . . . ⊕ Rdi ⊕ . . . ⊕ Rdr . Considerν = (ν1 , . . . , νd i ), where νk = νk for all k = j, and νj is defined by theequalitydi(μk + νk ) = li (c).k=1νjIf ≥ νj , then:= c+Π(μ, ν ) is a true coordinate parallelepiped.Note that P is a facet of Di+ (P ) unless ν = ν.If νj < νj , define μ = (μ1 , . . .

, μdi ) by setting μk = μk for allk = j, and μj = νj . Then Di− (P ) := c + Π(μ , ν) is a true coordinateparallelepiped, and P is a facet of Di− (P ). Let P be the facet of Di− (P )parallel to P .Di+ (P )71-06.tex : 2016/10/27 (14:27)page: 167Divided difference operators on polytopesWe now define Di (IP ) as follows:ID+ (P )iDi (IP ) =−ID− (P ) + IP + IP i167if νj ≤ νj ,if νj > νj .Remark 2.5. This definition is motivated by the following observation.

Let μ and ν be integers such that μ < ν. Define the functionf (μ, ν, t) of a complex variable t by the formulaf (μ, ν, t) = tμ + tμ+1 + . . . + tν ,that is, f is the exponential sum over all integer points in the segment[μ, ν] ⊂ R. Computing the sum of the geometric progression, we getthattμ − tν+1.f (μ, ν, t) =1−tThis formula gives a meromorphic continuation of f (μ, ν, t) to all real μand ν. In particular, for integer μ and ν such that μ > ν we obtainf (μ, ν, t) =tμ − tν+1= −(tν+1 + .

. . + tμ−1 ),1−tthat is, f is minus the exponential sum over all integer points in theopen interval (ν, μ) ⊂ R (cf. Example 2.4).Definition 3. Let P ⊂ Rd be a parapolytope such that P lies in thehyperplane {xij = const} for some j but does not lie in any hyperplane{xik = const} for k < j. Define Di (IP ) by setting(j)Di (IP ) c+Rdi = Di (IP ∩(c+Rdi ) )for all c in the complement to Rdi .

The superscript (j) on the right handside means that we always expand P ∩ (c + Rdi ) in the direction of xij asexplained above (even when P ∩ (c + Rdi ) for some c lies in a hyperplane{xik = const} for k < j).It is not hard to check that this definition yields a convex chain.In many cases (see examples in Section 3), Di (IP ) is the characteristicfunction of a polytope (and P is a facet of this polytope unless Di (IP ) =IP ).

This polytope will be denoted by Di (P ). The definition of Di canbe extended by linearity to the other parachains, however,Di (δ) for aconvex chain δ in general depends on a representation δ = P cP IP .The definition immediately implies that similarly to the classicalDemazure operators the convex-geometric ones satisfy the identity Di2 =Di . It would be interesting to find an analog of braid relations for theseoperators.71-06.tex : 2016/10/27 (14:27)page: 168168V.

Kiritchenko2.4. ExamplesDimension 2. The simplest meaningful example is R2 = R ⊕ R.Label coordinates in R2 by x := x11 and y := x21 . Assume that l1 = yand l2 = x. If P = {(μ1 , μ2 )} is a point, and μ2 ≥ 2μ1 , then D1 (P ) is asegment:D1 (P ) = [(μ1 , μ2 ), (μ2 − μ1 , μ2 )].If μ2 < 2μ1 , then D1 (IP ) is a virtual segment, that is,D1 (IP ) = −I[(μ2 −μ1 ,μ2 ),(μ1 ,μ2 )] + IP + I(μ2 −μ1 ,μ2 ) .If P = AB is a horizontal segment, where A = (μ1 , μ2 ) and B =(ν1 , μ2 ), then D2 (P ) is the trapezoid ABCD given by the inequalitiesμ1 ≤ x ≤ ν 1 ,μ2 ≤ y ≤ x − μ2 .See Figure 1 for D2 (P ) in the case μ1 = −1, ν1 = 2, μ2 = −1 (left) andμ1 = −1, ν1 = 2, μ2 = 0 (right). In the latter case, the convex chainD2 (IP ) is equal toIOBC − IADO + IOA + IDO − IO .Dimension 3.

A more interesting example is R3 = R2 ⊕ R. Labelcoordinates in R3 by x := x11 , y := x12 and z := x21 . Assume that l1 = zand l2 = x + y. If P = (μ1 , μ2 , μ3 ) is a point, then D1 (P ) is a segment:D1 (P ) = [(μ1 , μ2 , μ3 ), (μ3 − μ1 − 2μ2 , μ2 , μ3 )].Similarly, if P = [(μ1 , μ2 , μ3 ), (ν1 , μ2 , μ3 )] is a segment in R2 , then D1 (P )is the rectangle given by the equation z = μ3 and the inequalitiesμ1 ≤ x ≤ ν 1 ,μ2 ≤ y ≤ μ3 − μ1 − ν 1 − μ2 .Using the previous calculations, it is easy to show that if P =(λ2 , λ3 , λ3 ) is a point and λ3 < λ2 < −λ2 − λ3 , then D1 D2 D1 (P ) isthe 3-dimensional Gelfand–Zetlin polytope Qλ (as defined in Example2.1) for λ = (λ1 , λ2 , λ3 ), where λ1 = −λ2 − λ3 . Indeed, D2 D1 (P ) is thetrapezoid (see Figure 2) given by the equation y = λ3 and the inequalitiesλ2 ≤ x ≤ λ1 , λ3 ≤ z ≤ x.Then D1 D2 D1 (P ) is the union of all rectangles D2 (Ia ) for a ∈[λ3 , λ1 ], where Ia is the segment D2 D1 (P ) ∩ {z = a}, that is, Ia =[(max{z, λ2 }, λ3 , a), (λ1 , λ2 , a)].

Hence,λ3 ≤ y ≤ min{λ2 , z}.71-06.tex : 2016/10/27 (14:27)page: 169Divided difference operators on polytopes169Fig. 1. Trapezoids D2 (P ) for different segments P = AB.Similarly to the last example, we construct Gelfand–Zetlin polytopesfor arbitrary n using the string space from Example 2.1 (see Theorem3.4).71-06.tex : 2016/10/27 (14:27)170page: 170V. KiritchenkoFig. 2. Trapezoid D2 D1 (P ) and polytope D1 D2 D1 (P ) for apoint P = (0, −3, −3)71-06.tex : 2016/10/27 (14:27)page: 171Divided difference operators on polytopes§3.171Polytopes and Demazure characters3.1.

Characters of polytopesFor a string space Rd = Rd1 ⊕ . . . ⊕ Rdr , denote by σi (x) the sumof the coordinates of x ∈ Rd that correspond to the subspace Rdi , i.e.,diσi (x) = k=1xik . With each integer point x ∈ Rd in the string space,we associate the weight p(x) ∈ Rr defined as (σ1 (x), . . . , σr (x)). Forthe rest of the paper, we will always assume that li (x) depends onlyon p(x), that is, li comes from a linear function on Rr (the latter willalso be denoted by li ). In addition, we assume that li is integral, i.e.,li (x) ∈ Z for all x ∈ Zd .Denote the basis vectors in Rr by α1 , .

. . , αr , and denote the coordinates with respect to this basis by (y1 , . . . , yr ). For each i = 1, . . . r,define the affine reflection si : Rr → Rr by the formulasi (y1 , . . . , yi , . . . , yr ) = (y1 , . . . , li (y) − yi , . . . , yr ).Example 3.1. For the string space Rd = Rn−1 ⊕ Rn−2 ⊕ . . . ⊕ R1from Example 2.1, define the functions li by the formulali (x) = σi−1 (x) + σi+1 (x),where we put σ0 = σn = 0.

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