Диссертация (1136188), страница 59
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Eachpolytope has 11 vertices, hence, they are not combinatorially equivalentto string polytopes for s1 s2 s1 s2 or s2 s1 s2 s1 defined in [L].SL(3). Take G = SL(3) and w0 = s1 s2 s1 . The corresponding stringspace of rank 2 coincides with the one from Section 2.4, namely, R3 =R2 ⊕ R, and l1 = x21 , l2 = x11 + x12 . If aλ = (b, c, c) where −b − c ≥ b ≥ c,then the polytope D1 D2 D1 (aλ ) is the Gelfand–Zetlin polytope Qλ forλ = (−b − c, b, c). Label coordinates in R3 by x := x11 , y := x12 andz := x21 .
We now introduce a different structure of a string space in R3 bysplitting R2 , namely, R3 = R1 ⊕R1 ⊕R1 with coordinates x̃11 , x̃21 , x̃31 suchthat x̃11 = x, x̃21 = z, x̃31 = y. Put ˜l1 = l1 −2y, ˜l2 = l2 and ˜l3 = l1 −2x. Itis easy to check that D̃2 D̃1 (aλ ) = D2 D1 (aλ ) and deduce by argumentsof Example 3.3 that the virtual polytope D̃3 D̃2 D̃1 (aλ ) (see Figure 4)has the same character as the polytope D1 D2 D1 (aλ ). In particular,the image of D̃3 D̃2 D̃1 (aλ ) under the projection (x, y, z) → (x + y, z)coincides with the weight polytope of the irreducible representation ofSL3 with the highest weight −cα1 − (a + b)α2 (provided that the latteris dominant, that is, a + b − 2c ≥ 0 and c − 2(a + b) ≥ 0).
The virtualpolytope D̃3 D̃2 D̃1 (aλ ) is a twisted cube of Grossberg–Karshon (cf. [GK,Figure 2]) given by the inequalitiesy − b ≤ z − c + λ2 ,a ≤ x ≤ c − 2b − a,0 ≤ t − d ≤ λ2 ,c ≤ z ≤ x + b − c,t−d≤b ≤ y ≤ −2x + z − b. (GK)Note that the last pair of inequalities is inconsistent when b > −2x +z − b, and should be interpreted in the sense of convex chains.
Moreprecisely, D̃3 D̃2 D̃1 (aλ )=IP − IQ , where P is the convex polytope givenby inequalities (GK) and Q is the set given by the inequalitiesa ≤ x ≤ c − 2b − a,c ≤ z ≤ x + b − c,b > y > −2x + z − b.(cf. [GK, Formula (2.21)]). A generalization of this example will begiven in Section 4.3.71-06.tex : 2016/10/27 (14:27)178page: 178V. KiritchenkoFig. 4.
Virtual polytope D̃3 D̃2 D̃1 (aλ ) for aλ = (0, −3, −3)§4.Bott towers and Bott–Samelson resolutionsIn this section, we outline possible algebro-geometric applications ofthe convex-geometric Demazure operators.4.1. Bott towersLet us recall the definition of a Bott tower (see [GK] for more details). It is a toric variety obtained from a point by iterating the followingstep. Let X be a toric variety, and L a line bundle on X. Define a newtoric variety Y := P(L ⊕ OX ) as the projectivization of the split ranktwo vector bundle L ⊕ OX on X. Consider a sequence of toric varietiesY0 ← Y1 ← . . . ← Yd ,where Y0 is a point, and Yi = P(Li−1 ⊕ OYi−1 ) for a line bundle Li−1 onYi−1 .
In particular, Y1 = P1 and Y2 = P(OP1 ⊕ OP1 (k)) is a Hirzebruchsurface. We call Yd the Bott tower corresponding to the collection of linebundles (L1 , . . . , Ld−1 ). Note that the collection (L1 , . . . , Ld−1 ) dependsinteger parameters since Pic(Yi ) = Zi . Recall that the Picardon d(d−1)2group of a toric variety of dimension d can be identified with a group71-06.tex : 2016/10/27 (14:27)page: 179Divided difference operators on polytopes179of virtual lattice polytopes in Rd in such a way that very ample linebundles get identified with their Newton polytopes. One can describethe (possibly virtual) polytope P (L) of a given line bundle L on Yd usinga suitable string space.Consider a string space with d = r, that is, d1 = .
. . = dr = 1. Wehave the decompositionRd = R ⊕ . . . ⊕ R .dLabel coordinates in Rd as follows: xi1 := yi for i = 1,. . . , d. Sincewe will be interested in the polytope P := D1 . . . Dd (a), we can assumethat the linear function li for i < d does not depend on y1 , . . . , yi ,and ld = y1 . Hence, the collection (l1 , . .
. , ld−1 ) of linear functions alsoparameters.depends on d(d−1)2The projective bundle formula gives a natural basis (η1 , . . . , ηd ) inthe Picard group of Yd . Namely, for d = 1, the basis in Pic(P1 ) consistsof the class of a point in P1 . We now proceed by induction. Let (η1 ,. .
. , ηi−1 ) be the basis in Pic(Yi−1 ) (we identify Pic(Yi−1 ) with its pullback to Pic(Yd )). Put ηi = c1 (OYi (1)) where c1 (OYi (1)) denotes thefirst Chern class of the tautological quotient line bundle OYi (1) on Yi .Decompose L1 ,. . . , Ld−1 in the basis (η1 , . . . , ηd ):L1 = a1,1 η1 ,...,Ld−1 = ad−1,1 η1 + . . . + ad−1,d−1 ηd−1 .(∗)Similarly, decompose l1 ,. . . , ld−1 in the basis of coordinate functions(y1 , . . . , yd ):l1 = b1,1 y2 + . . . + b1,d−1 yd ,...,ld−1 = bd−1,d−1 yd .(∗∗).Let Yd be the Bott tower corresponding to the collection (∗) of linebundles, and Rd the string space corresponding to the collection (∗∗) oflinear functions.
One can show (cf. [GK, Theorem 3]) that if ai,j = bj,i ,then there exists aL ∈ Rd such thatP (L) = D1 D2 . . . Dd (aL ).In particular, when L is very ample the polytope D1 D2 . . . Dd (aL )coincides with the Newton polytope of the pair (Yd , L). Note that theintermediate polytopes {aL } ⊂ Dd (aL ) ⊂ Dd−1 Dd (aL ) ⊂ . . . correspondto the flag of toric subvarieties Z0 = {pt} ⊂ Z1 ⊂ .
. . ⊂ Zd = Yd , whereZi = p−1d−i (Z0 ) and pi is the projection pi : Yd → Yi .71-06.tex : 2016/10/27 (14:27)180page: 180V. KiritchenkoFig. 5. Polytope D1 Eu D2 D1 (a) for a = (0, −1, −1) and u =(0, −1/2, 0)4.2. Bott–Samelson varieties.Similarly to Bott towers, Bott–Samelson varieties can be obtainedby successive projectivizations of rank two vector bundles. In general,these bundles are no longer split, so the resulting varieties are not toric.In [GK], Bott–Samelson varieties were degenerated to Bott towers bychanging complex structure (in particular, Bott–Samelson varieties arediffeomorphic to Bott towers when regarded as real manifolds).
Belowwe define these varieties using notation of Section 3.3.Fix a Borel subgroup B ⊂ G. With every collection of simple roots(αi1 , . . . , αi ), one can associate a Bott–Samelson variety R(i1 ,...,i ) anda map R(i1 ,...,i ) → G/B by the following inductive procedure. PutR∅ = pt. For every -tuple I = (i1 , . . . , i ) denote by I j the ( − 1)tuple (i1 , . .
. , îj , . . . , i ). Define RI as the fiber product RI ×G/P G/B,where Pi is the minimal parabolic subgroup corresponding to the rootαi . The map rI : RI → G/B is defined as the projection to the secondfactor. There is a natural embeddingRI → RI ;x → (x, rI (x)).In particular, any subsequence J ⊂ I yields the embedding RJ → RI .71-06.tex : 2016/10/27 (14:27)page: 181Divided difference operators on polytopes181It follows from the projective bundle formula that the Picard groupof RI is freely generated by the divisors RI 1 ,. . .
, RI . Denote by vthe geometric valuation on C(RI ) defined by the flag R∅ ⊂ R(i ) ⊂R(i−1 ,i ) ⊂ . . . ⊂ RI 1 ⊂ RI . Let L be a line bundle on RI , and Pv (L)its Okounkov body with respect to the valuation v. Conjecturally, Pv (L)can be described using string spaces as follows.Replace a reduced decomposition of w0 in the definition of the stringspace from Section 3.3 by a sequence (αi1 , .
. . , αi ) that defines RI (weno longer require that si1 . . . si be reduced). More precisely, let di thenumber of αij in this sequence such that ij = i. We get the followingstring space SI of rank ≤ r and dimension :R = Rd1 ⊕ . . . ⊕ Rdr ,where the functions li are given by the formula:(αk , αi )σk (x).li (x) =(BS)k=iIn particular, if = d and si1 · · · si is reduced then RI is a Bott–Samelson resolution of the flag variety G/B, and R is exactly the stringspace from Section 3.3. Denote by Eu the parallel translation in thestring space by a vector u ∈ R .Conjecture 4.1.
For every line bundle L on RI , there exists apoint μ ∈ Rr and vectors u1 , . . . , u ∈ R such that we havePv (L) = Eu1 Di1 Eu2 Di2 . . . Eu Di (aμ )for any point aμ ∈ R that satisfies p(a) = μ.In particular, if L = rI∗ L(λ), where L(λ) is the line bundle on G/Bcorresponding to the dominant weight λ, then one can take u2 = . . . =u = 0 and μ = λ. This conjecture agrees with the example computedin [Anderson, Section 6.4] for SL3 and the Bott–Samelson resolutionR(1,2,1) (cf. Figure 5 and Figure 3(b) in loc.cit.).
Figure 5 shows thepolytope D1 Eu D2 D1 (a) for the string space (BS) when G = SL3 andI = (1, 2, 1).4.3. Degenerations of string spacesWhile twisted cubes of Grossberg–Karshon for GLn and Gelfand–Zetlin polytopes have different combinatorics they produce the sameDemazure characters. We now reproduce this phenomenon for generalstring spaces.
In particular, we transform the string space (BS) fromSection 4.2 into a string space from Section 4.1.71-06.tex : 2016/10/27 (14:27)page: 182182V. KiritchenkoLet S be a string space Rd = Rd1 ⊕ . . . ⊕ Rdr with functions l1 ,. . . ,lr . Suppose that di > 1.Definition 4. The i-th degeneration of the string space S is the stringspaceRd = Rd1 ⊕ . . . ⊕ Rdi −1 ⊕ R1 ⊕ . . .
⊕ Rdrof rank (r + 1) with functions l1 ,. . . ,li (x) = li (x) − 2xidi ;Rd ili , li ,. . . , lr ,whereli (x) = li (x) − 2di −1xik .k=1Example 4.2. The string space R ⊕ R ⊕ R from Example 3.4 is the1-st degeneration of the space R2 ⊕ R with l1 = x11 + x12 , l2 = x21 .: Rr+1→ Rr by sendingDefine the projection pi (y1 , . . . , yi , yi , . . . , yr ) to (y1 , . .
. , yi + yi , . . . , yr ). This projection induces a homomorphism of group algebras of the lattices Zr+1 and Zr ,which we will also denote by pi . It is easy to check thatTi ◦ pi = pi ◦ Ti = pi ◦ Ti .Combining this observation with Theorem 3.2, we get the followingproposition.Proposition 4.3. For a lattice polytope P ⊂ Rd , we haveχ(Di (P )) = pi (χ(Di (P ))) = pi (χ(Di (P ))).We now degenerate successively the string spaces from Section 4.2.Let I = (αi1 , . . .
, αi ) be a sequence of simple roots, andR = Rd1 ⊕ . . . ⊕ Rdris the corresponding string space SI with the functions l1 ,. . . , lr givenby (BS).Let S˜I be the string space of rank obtained from SI by (d1 − 1)first degenerations, (d2 − 1) second degenerations etc., that is,(1)(1)()()S˜I = R1 ⊕ . . . ⊕ Rd1 ⊕ . . . ⊕ R1 ⊕ .
. . ⊕ Rd ,Rd 1Rd (i)where the functions lj are given by the formula:(i)xik .lj (x) = li (x) − 2k=j71-06.tex : 2016/10/27 (14:27)page: 183Divided difference operators on polytopes183(i)Denote the Demazure operators associated with S˜I by D̃j .For a point a ∈ R , consider the convex chain PI = Di1 . . . Di (a).For every k = 1,. . . , r, we now replace the rightmost Dk in the expression(k)(k)(k)Di1 . . . Di by D1 , the next one by D2 ,. . . , the leftmost by Dik .Denote the resulting convex chain by P̃I .Example 4.4.
If r = 3, = 6 and I = (α1 , α2 , α1 , α3 , α2 , α1 ), thenPI = D1 D2 D1 D3 D2 D1 ,(1)(2)(1)(3)(2)(1)P̃I = D̃3 D̃2 D̃2 D̃1 D̃1 D̃1 (a).Proposition 4.3 implies that PI and P̃I have the same character(with respect to the map p : x → σ1 (x)α1 + . . . + σr (x)αr ).Corollary 4.5. If p(a) is dominant and si1 · · · si is reduced, thenthe corresponding Demazure character coincides with χ(PI ) = χ(P̃I ).The proof is completely analogous to the proof of Theorem 3.6.Remark 4.6. Note that the convex chain P̃I coincides with thetwisted cube constructed in [GK] for the corresponding Bott–Samelsonresolution. Indeed, (αi , αi ) = −2 according to our definition of thefunction (·, αi ) (see Section 3.3), hence, we can write(i)lj (x) =(αp , αi )xpq .(p,q)=(i,j)It is now easy to check that the defining inequalities for P̃I coincide withthe inequalities given by [GK, Formula (2.21)] together with computations of [GK, Section 3.7].The string space SI and its complete degeneration S˜I are two extreme cases that yield convex chains for given Demazure characters.
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