Диссертация (1136188), страница 58
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Identify Rn−1 with the weight lattice ofSLn so that αi is identified with the i-th simple root. In this case,the reflection si coincides with the simple reflection in the hyperplaneperpendicular to the root αi .We now consider the ring R of Laurent polynomials in the formalexponentials t1 := eα1 , . . . , tr := eαr (that is, R is the group algebraof the lattice Zr ⊂ Rr ). Let P ⊂ Rd be a lattice polytope in the stringspace, i.e., the vertices of P belong to Zr . Define the character of P asthe sum of formal exponentials ep(x) over all integer points x inside andat the boundary of P :χ(P ) :=ep(x) .x∈P ∩ZdIn particular, if d = r, then χ(P ) is exactly the integer point transformof P .
The R-valued function χ can be extended by linearity to all latticeconvex chains, that is, to the chains P cP IP such that P is a latticepolytope and cP ∈ Z.Define the Demazure operator Ti on R as follows:Ti f =f − ti · s i f,1 − ti71-06.tex : 2016/10/27 (14:27)page: 172172V. Kiritchenkowhere the action of si on R is defined by si eλ := esi (λ) for λ ∈ Zr . Forthe string space of Example 3.1, these operators reduce to the classicalDemazure operators on the group algebra of the weight lattice of SLn .The following result motivates the definition of divided differenceoperators Di on convex chains (Definition 3).Theorem 3.2. Let P ⊂ Rd be a lattice parapolytope.
Thenχ(Di (IP )) = Ti χ(P ).Proof. By definition of Di (IP ), it suffices to prove this identitywhen P = c + Γ, where c lies in the complement to Rdi and Γ :=Π(μ, ν) ⊂ Rdi is a coordinate parallelepiped. Then σ(z)ti .χ(P ) = ep(c)z∈Γ∩ZdiHence,⎛Ti (χ(P )) = ep(c) Ti ⎝⎞σ(z) ⎠ti.z∈Γ∩ZdiRecall that by definition of ν we havedi(μk + νk ) = li (c).k=1Assume that νj ≥ νj . Let Π denote Π(μ, ν ). Then Γ, Π and Ti satisfythe hypothesis of [KST, Proposition 6.3]. Applying this proposition weget that⎛⎞ σ(z) σ(z)ti ⎠ =ti .Ti ⎝z∈Γ∩Zdiz∈Π∩ZdiHence, Ti (χ(P )) = χ(Di (P )).The case νj < νj is completely analogous.Q.E.D.Note that Theorem 3.2 for di = 1 follows directly from the definitionsof Ti and Di (see Remark 2.5).Example 3.3. Figure 3 illustrates Theorem 3.2 when di = 2 andP = c + Γ where Γ ⊂ Rdi is the segment [(−1, −1), (2, −1)].
Namely,xi +xiTi (ti 1 2 ) is equal to the character of the segment [(xi1 , xi2 ), (xi1 , li (c) −2xi1 − xi2 )] for every (xi1 , xi2 ) ∈ Γ ∩ Z2 by definition of Ti . Hence,xi +xiTi (ti 1 2 )(xi1 ,xi2 )∈Γ∩Z271-06.tex : 2016/10/27 (14:27)page: 173Divided difference operators on polytopes173Fig. 3. Rectangle and trapezoid yield the same characterfor li (c) = 3 coincides with the character of the trapezoid shown onFigure 3 (left). It is easy to construct a bijective correspondence betweenthe integer points in the trapezoid and those in the rectangle Di (P ) insuch a way that the sum of coordinates is preserved.
The former aremarked by black dots, and the latter by empty circles.Theorem 3.2 allows one to construct various polytopes (possibly virtual) and convex chains whose characters yield the Demazure characters(in particular, the Weyl character) of irreducible representations of reductive groups (see Section 3.3). The same character can be capturedusing string spaces for different partitions d = d1 + d2 + .
. . + dn (seeSection 4.3). The case d1 = . . . = dn = 1 produces polytopes with verysimple combinatorics, namely, multidimensional versions of trapezoidsthat are combinatorially equivalent to cubes (they are called twistedcubes in [GK]). However, twisted cubes that represent the Weyl characters are virtual.
Considering string spaces with di > 1 allows one torepresent the Weyl character by a true though more intricate polytope(see Example 3.4 for SL3 and Section 4.3). The reason is illustrated byFigure 3 (right) that depicts a virtual trapezoid and a (true) rectangle71-06.tex : 2016/10/27 (14:27)page: 174174V.
Kiritchenkowith the same character. Note that the point (2, 1) (marked by −1)contributes a negative summand to the character of the trapezoid.3.2. Gelfand–Zetlin polytopes for SLnLet Rd = Rn−1 ⊕ Rn−2 ⊕ . . . ⊕ R1 be the string space of rank (n − 1)from Example 3.1. The theorem below shows how to construct theclassical Gelfand–Zetlin polytopes (see Example 2.1) via the convexgeometric Demazure operators D1 ,.
. . , Dn−1 .Theorem 3.4. For every strictly dominant weight λ = (λ1 , . . . , λn )(that is, λ1 > . . . > λn ) of GLn such that λ1 + . . . + λn = 0, theGelfand–Zetlin polytope Qλ coincides with the polytope[(D1 )(D2 D1 )(D3 D2 D1 ) . . . (Dn−1 . .
. D1 )] (aλ ),where aλ ∈ Rd is the point (λ2 , . . . , λn ; λ3 , . . . , λn ; . . . ; λn ).Proof.Let us define the polytopePλ (i, j) := (D̂n−j . . . D̂i Di−1 . . . D1 ) . . . (Dn−1 . . . D1 ) (aλ )for every pair (i, j) such that 1 ≤ i ≤ (n − j) ≤ (n − 1). Put x0l = λifor l = 1,. . . , n. We will show by induction on dimension that Pλ (i, j)is the face of the Gelfand–Zetlin polytope Qλ given by the equationsxkl = xk−1l+1 for all pairs (k, l) such that either l > j, or l = j and k ≥ i.The induction base is Pλ (1, 1) = aλ , which is clearly a vertex of Qλby our assumption. The induction step follows from Lemma 3.5 below.Hence, Pλ (1, n − 1) is the facet of Qλ given by the equation x1n−1 = λn .Applying Lemma 3.5 again, we get that D1 (Pλ (1, n−1)) = Qλ . Q.E.D.Note that any Gelfand–Zetlin polytope Qλ can be obtained by aparallel translation from one with λ1 + .
. . + λn = 0.The lemma below can be easily deduced directly from the definitionof Di using Example 2.1 together with an evident observation that a+b =min{a, b} + max{a, b} for any a, b ∈ R.Lemma 3.5. Let Γ be a face of the Gelfand–Zetlin polytope Qλgiven by the following equationsxi−11xi−12xi−13...xi−1jxi1xi2...xij−1xi−1j+1xijxi+11...xi+1j−2xi+1j−1xi−1j+2xij+1xi+1j.........xi−1n−i+1xin−i.xi+1n−i−171-06.tex : 2016/10/27 (14:27)page: 175Divided difference operators on polytopes175as well as by (possibly) other equations that do not involve variablesxi1 ,. .
. , xin−i . Then the defining equations of Di (Γ) are obtained fromthose of Γ by removing the equation xij = xi−1j+1 .Recall that integer points inside and at the boundary of the Gelfand–Zetlin polytope Qλ by definition of this polytope parameterize a natural basis (Gelfand–Zetlin basis) in the irreducible representation ofGLn with the highest weight λ. Under this correspondence, the mapp : Rd → Rn−1 assigns to every integer point the weight of the corresponding basis vector. Combining Theorem 3.4 with Theorem 3.2 onegets a combinatorial proof of the Demazure character formula for the decomposition w0 = (s1 )(s2 s1 )(s3 s2 s1 ) .
. . (sn−1 sn−2 . . . s1 ) of the longestword in Sn (in this case, the Demazure character coincides with theWeyl character of Vλ ). Here si denotes the elementary transposition(i, i + 1) ∈ Sn .3.3. Applications to arbitrary reductive groupsWe now generalize Gelfand–Zetlin polytopes to other reductivegroups using Theorem 3.2. Let G be a connected reductive group ofsemisimple rank r. Let α1 ,. . .
, αr denote simple roots of G, and s1 ,. . . ,sr the corresponding simple reflections. Fix a reduced decompositionw0 = si1 si2 · · · sid where w0 is the longest element of the Weyl groupof G. Note that d is the length of w0 , which is equal to the number ofpositive roots as well as to the dimension of the complete flag variety ofG. Let di be the number of sij in this decomposition such that ij = i.Consider the string spaceRd = Rd1 ⊕ . .
. ⊕ Rdr ,where the functions li are given by the formula:(αk , αi )σk (x)li (x) =k=idi ixj ). Here (αk , αi ) is determined by the simple(recall that σi (x) = j=1reflection si as follows:si (αk ) = αk + (αk , αi )αi ,(that is, the function (·, αi ) is minus the coroot corresponding to αi ).In particular, if G = SLn and w0 = (s1 )(s2 s1 )(s3 s2 s1 ) . . . (sn−1 . . . s1 ),then we get the string space from Example 3.1.Define the projection p of the string space to the real span Rr of theweight lattice of G by the formula p(x) = σ1 (x)α1 + . .
. + σr (x)αr . Note71-06.tex : 2016/10/27 (14:27)176page: 176V. Kiritchenkothat after identifying the root lattice of G with Zr ⊂ Rr the map p isthe same as the map p defined earlier in Section 3.1.Theorem 3.6. For every dominant weight λ in the root lattice ofG, and every point aλ ∈ Zd such that p(aλ ) = w0 λ the convex chainPλ := Di1 Di2 .
. . Did (aλ )yields the Weyl character χ(Vλ ) of the irreducible G-module Vλ , that is,χ(Vλ ) = χ(Pλ ).Proof.By the Demazure character formula [Andersen] we haveχ(Vλ ) = Ti1 . . . Tid ew0 λ .This formula together with Theorem 3.2 implies by induction the desiredstatement.Q.E.D.As a corollary, we get that p∗ (Pλ ) is the weight polytope of Vλin Rr . Here p∗ denotes the push-forward of convex chains (see [PKh,Proposition-Definition 2]).Remark 3.7. A slight modification of Theorem 3.2 makes it applicable to all dominant weights (not only those inside the root lattice).Namely, instead of the lattices Zd ⊂ Rd and Zr ⊂ Rr one should consider the shifted lattices aλ + Zd ⊂ Rd and λ + Zr ⊂ Rr , and definecharacters of polytopes with respect to these new lattices.
The convexchain Pλ will be lattice with respect to the lattice aλ + Zd .In the same way, we can construct convex chains that capture thecharacters of Demazure submodules of Vλ for any element w in the Weylgroup and a reduced decomposition w = sj1 . . . sj (see Corollary 4.5).In particular, if sj1 . . . sj is a terminal subword of si1 si2 · · · sid (that is,j = id , j−1 = id−1 , etc.) then the corresponding convex chain is aface of Pλ . It is interesting to check whether this convex chain is alwaysa true polytope. One way to do this would be to identify it with anOkounkov polytope of the Bott–Samelson resolution corresponding tothe word sj1 .
. . sj (see Conjecture 4.1).3.4. ExamplesSp(4). Take G = Sp(4) and w0 = s2 s1 s2 s1 (here α1 denotes theshorter root and α2 denotes the longer one). The corresponding stringspace of rank 2 is R4 = R2 ⊕ R2 together with l1 = 2(x21 + x22 ) andl2 = x11 + x12 . Let λ = −p1 α1 − p2 α2 be a dominant weight, that is, λ1 :=(p2 − p1 ) ≥ 0 and λ2 := (p1 − 2p2 ) ≥ 0. Choose a point aλ = (a, b, c, d)71-06.tex : 2016/10/27 (14:27)page: 177Divided difference operators on polytopes177such that (a + b) = p1 and (c + d) = p2 (that is, p(aλ ) = w0 λ = −λ).Label coordinates in R4 by x := x11 , y := x12 , z := x21 and t := x22 . Thenthe polytope D2 D1 D2 D1 (aλ ) is given by inequalities0 ≤ x − a ≤ 2λ1 ,z − c ≤ x − a + λ2 ,y − b ≤ 2(z − c),y−b.2It is not hard to show that the polytopes D1 D2 D1 D2 (aλ ) andD2 D1 D2 D1 (aλ ) are the same up to a linear transformation of R4 .
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