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We were inspired by the results of Bressler–Evens on Schubertcalculus in complex cobordism [BE], and part of our motivation was to transfer theirresults to the algebro-geometric setting. We also deduced a cobordism version ofChevalley–Pieri formula.There are several major differences between Chow ring/cohomology and K-theoryof flag varieties on the one hand, and algebraic/complex cobordism on the otherhand. For instance, classes of Schubert varieties provide a natural basis in theformer theories but not in the latter ones since Schubert varieties are in general notsmooth. Instead, Bott–Samelson resolutions of Schubert varieties form a naturalgenerating set (but not a basis) for the ring of algebraic/complex cobordism.In [6] (joint work with Amalendu Krishna), we describe equivariant algebraiccobordism rings of flag varieties and wonderful compactifications of symmetric spacesof minimal rank.
In particular, spaces (G×G)/Gdiag are symmetric of minimal rank.In the case of flag varieties, we used complex cobordism and topological arguments.Recently, a purely algebro-geometric approach was proposed in [CZZ]. In the caseof wonderful compactifications, we used approach of Brion–Joshua who describedequivariant Chow rings [BJ].In [9], analogs of Demazure operators on convex polytopes are defined. Such operators in general take a polytope P to a polytope or a convex chain P 0 of dimensionone greater so that the exponential sums over the lattice points in P and P 0 are related by the classical Demazure operator.
In particular, convex geometric Demazureoperators can be applied inductively to construct the GZ polytopes in type A andKarshon–Grossberg cubes in any type from a single point. In type C2 , they wereused to construct symplectic DDO polytopes that are combinatorially different bothfrom symplectic GZ and FFLV polytopes. It turned out that these polytopes coincide with Newton–Okounkov polytopes of the symplectic flag variety for a naturalgeometric valuation [8]. Recently, Naoki Fujita conjectered that several classes ofDDO polytopes coincide with Nakashima–Zelevinsky polyhedral realizations (for C2this follows from [FN, Example 5.10]).In [10], Newton–Okounkov polytopes of complete flag varieties in type A arecomputed for a geometric valuation given by a flag of translated Schubert subvarieties that correspond to terminal subwords in the longest word decomposition(s1 )(s2 s1 )(s3 s2 s1 )(.
. .)(sn−1 . . . s1 ) (Examples 1.4, 1.7, 2.3 illustrate this computationfor n = 3, see also Section 4.3). Surprisingly, the resulting polytopes turned outto coincide with the FFLV polytopes though the latter were originally constructedusing a different approach. This coincidence stimulated further research (see [FaFL]for more details).10List of publications[1] V. Kiritchenko, Chern classes of reductive groups and an adjunction formula, Ann. Inst.Fourier 56 (2006), no.
4, 1225–1256[2] —, On intersection indices of subvarieties in reductive groups, Moscow Math. J., 7 (2007),no. 3 (issue dedicated to Askold Khovanskii), 489–505[3] —, Gelfand-Zetlin polytopes and flag varieties, IMRN (2010), no. 13, 2512–2531[4] J. Hornbostel, —, Schubert calculus for algebraic cobordism, J. reine angew. Math.(Crelles), 2011, no. 656, 59–85[5] —, E. Smirnov, V. Timorin, Schubert calculus and Gelfand–Zetlin polytopes, RussianMath. Surveys 67 (2012), no.
4, 685–719[6] —, A. Krishna, Equivariant Cobordism of Flag Varieties and of Symmetric Varieties,Transform. Groups, 18 (2013), no. 2, 391–413[7] P. Gusev, —, V. Timorin, Counting vertices in the Gelfand-Zetlin polytopes, J. of Comb.Theory, Series A, 120 (2013), 960–969[8] —, Geometric mitosis, Math. Res. Lett., 23 (2016), no. 4, 1069–1096[9] —, Divided difference operators on polytopes, Adv.
Studies in Pure Math. 71 (2016), 161–184[10] —, Newton–Okounkov polytopes of flag varieties, Transform. Groups 22 (2017), no. 2,387–4024. Main resultsThis section includes more detailed statements of the main results of the habilitation thesis. We try to make formulations as self-contained as possible. However, weassume that the reader is familiar with representation theory and Schubert calculus.4.1. Euler characteristic of complete intersections in reductive groups.Let G be a connected complex reductive group of dimension n and rank k, and letπ : G → GL(V ) be a faithful representation of G.
A generic hyperplane section Hπcorresponding to π is the preimage π −1 (H) of the intersection of π(G) with a genericaffine hyperplane H ⊂ End(V ). It is not hard to show that all generic hyperplanehave the same (topological) Euler characteristic. Below we give an explicit formulafor the Euler characteristic of Hπ .
It follows from [1, Theorem 1.1], [2, Theorem1.3], which also imply an analogous formula for the Euler characteristic of completeintersections.Choose a maximal torus T ⊂ G, and denote by LT its character lattice. Choosealso a Weyl chamber D ⊂ LT ⊗ R. Denote by R+ the set of all positive roots of Gand denote by ρ the half of the sum of all positive roots of G.
The inner product(·, ·) on LT ⊗ R is given by a nondegenerate symmetric bilinear form on the Liealgebra of G that is invariant under the adjoint action of G (such a form exists sinceG is reductive). Let Pπ ⊂ LT ⊗ R denote the weight polytopes of the representationπ, that is, the convex hull of the weights of T that occur in π.Define a polynomial function F (x, y) on (LT ⊕ LT ) ⊗ R by the formula:Y (x, α)(y, α).F (x, y) =(ρ, α)2+α∈R11From a geometric viewpoint, this polynomial counts the self-intersection indices ofdivisors on the product G/B × G/B of two flag varieties (divisors correspond to theweights (λ1 , λ2 ) ∈ LT ⊕ LT ).Theorem 4.1. [1,Theorem 1.1], [2,Theorem 1.2] Let D be the differential operator(on functions on (LT ⊕ LT ) ⊗ R) given by the formulaYD=(1 + ∂α )(1 + ∂eα ),α∈R+where ∂α and ∂eα are directional derivatives along the vectors (α, 0) and (0, α), respectively. Denote by [D]i the i-th degree term in D (if D is regarded as polynomialin ∂α and ∂eα ).
ThenZn−1χ(Hπ ) = (−1)(n! − (n − 1)![D]1 + (n − 2)![D]2 − . . . + k![D]n−k ) F (x, x)dx.Pπ ∩DThe volume form dx is normalized so that the covolume of the lattice LT in LT ⊗ Ris equal to 1.For instance, if G = SL3 (C) and π is an irreducible representation with the highestweight mω1 + nω2 (by ω1 and ω2 we denote fundamental weights), then we get thefollowing answer:χ(Hπ ) = −3(m8 + 16m7 n + 112m6 n2 + 448m5 n3 + 700m4 n4 + 448m3 n5 + 112m2 n6 +16mn7 + n8 + 18(m6 + 12m5 n + 50m4 n2 + 80m3 n3 + 50m2 n4 + 12mn5 + n6 )++6(5m4 + 40m3 n + 72m2 n2 + 40mn3 + 5n4 ) + 6(m2 + 4mn + n2 )−−6(m + n)(m6 + 13m5 n + 71m4 n2 + 139m3 n3 + 71m2 n4 + 13mn5 + n6 ++5(m4 + 9m3 n + 19m2 n2 + 9mn3 + n4 ) + 3(m2 + 5mn + n2 ))).4.2. Convex geometric models for Schubert calculus.
In [3,5] Gelfand–Zetlinpolytope is used to model Schubert calculus on the variety of complete flags in Cn .Intersection product of cyles on the flag variety corresponds to the intersection offaces of the polytope. For an arbitrary reductive group G, we can also constructmodels for Schubert calculus on the flag variety G/B using polytopes. Recall thatthe Chow ring CH ∗ (G/B) (regarded as a group under addition) is a free Abeliangroup with the basis od Schubert cycles [Xw ], which are labeled by elements w ∈ Wof the Weyl group of G. To construct a useful model we need to find out which linearcombinations of faces of a polytope correspond to Schubert cycles. In the case ofG = GLn (C), we achieved this goal using the combinatorial mitosis of Knutson–Miller.
However, there were no suitable algorithms for the other groups. In [8,9],such algorithms are developed for an arbitrary G (in particular, for G = GLn (C) weget the Knutson–Miller mitosis).Fix a reduced decomposition of the longest element w0 ∈ W (by si we denotesimple reflections): w0 = si1 .
. . sid (that is, d = dim G/B). Convex geometric12Demazure operators Di were constructed in [9] be elementary methods. Namely,with every simple reflection si we associate an operation on polytopes in Rd thatraises dimension by one. In particular, using the decomposition w0 and a dominantweight λ we can construct inductively a polytope (possibly virtual) that encodesthe Weyl character. First, we define a linear operator p : Rd → Rk that associatesweights of G with the lattice points in Rd (recall that k denotes the rank of G).Theorem 4.2. [9,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,Xχ(Vλ ) =ep(x) .x∈Pλ ∩ZdIt is not hard to check that for every element w ∈ W there exists a reduceddecomposition w = si1 . . . si` such that w is a subword of w0 . In [9], for everysimple reflection si we construct an operation Mi (geometric mitosis) on faces of thepolytope Pλ . In particular, under additional assumptions we obtain by inductionon ` a collection of faces that encode the Demazure character χw (λ) correspondingto the Schubert variety Xw and the weight λ:Theorem 4.3. [8, Corollary 3.6] Let Pλ ⊂ Rd be an admissible λ-balanced parapolytope, and Sw ⊂ Pλ the union of all faces produced from the vertex 0 ∈ Pλ byapplying successively the operations Mj` ,.
. . , Mj1 . Suppose that for every 1 < k ≤ `,the collection of faces Mjk . . . Mj` (0) satisfies conditions (3) and (4) of [8, Theorem3.4]. ThenXχw0 w (λ) = ew0 λep(x) .x∈Sw ∩ZdFor instance, for G = Sp4 (C) and w0 = s2 s1 s2 s1 we get a polytope Pλ ⊂ R4that can also be obtained as the Newton–Okounkov polytope of the flag varietyX = Sp4 /B and the line bundle Lλ for the valuation associated with the flags2 s1 s2 s1 Xid ⊂ s2 s1 s2 Xs1 ⊂ s2 s1 Xs2 s1 ⊂ s2 Xs1 s2 s1 ⊂ X of translated Schubertsubvarieties [8, Proposition 4.1].
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