Диссертация (1136188), страница 47
Текст из файла (страница 47)
Put m = −(χ, α). We can express xχ − xsα χ = xχ − xχ+mα as a formalpower series H(x, y) ∈ Lgr [[x, y]] in x = xχ and y = xα using the universal formalgroup law. Then H(x, y) is homogeneous and divisible by y [29, (2.5.1)] so thatthe ratio H(x, y)/y is a homogeneous power series. In particular, xχ − sα (xχ ) isdivisible by xα .Next, note that if the lemma holds for f and g, then it also holds for f g, sincef g − sα (f g) = (f − sα (f ))g + sα (f )(g − sα (g)). In particular, the lemma holds forany monomial in xχ as desired.
Theorem 6.4. Let X be a wonderful symmetric variety of minimal rank. Thenthe composite map∗∗sGT : ΩG (X) → (ΩT (X))W→ (Ω∗T (X))WKis a ring isomorphism with the rational coefficients.→ (Ω∗T (Y ))WK(6.4)410VALENTINA KIRITCHENKO, AMALENDU KRISHNAProof. All the arrows in (6.4) are canonical ring homomorphisms. The isomorphism of the first arrow follows from [22, Thm. 8.6]. We recall here that the proofof [22, Thm. 8.6] is based on a spectral sequence of Hopkins–Morel for the motivic cobordism.
Although the result of Hopkins–Morel has not been published yet(however, see [19]), the rational version of their spectral sequence and its degeneration is known and is an immediate consequence of [32, Cor. 10.6(ii)].Once we know the first isomorphism in (6.4), it suffices to show that the map(Ω∗T (X))W → (Ω∗T (Y ))WK is an isomorphism. We prove this by adapting theargument of [7, Thm. 2.2.1].Since X has only finitely many T -fixed points and finitely many T -stable curves,it follows from [23, Thm. 7.8] and Lemma 6.2 that Ω∗T (X) is isomorphic as an Salgebra to the space of tuples (fw·z )w∈W/WL of elements of S such thatfv·z ≡ fw·z (mod xχ )whenever v ·z and w ·z lie in an irreducible T -stable curve on which T acts throughits character χ.
Under this isomorphism, the ring S is identified with the constanttuples (f ).We deduce from this that (Ω∗T (X))W is isomorphic, via the restriction to theT -fixed point z, to the subring of S WL consisting of those f such thatv −1 (f ) ≡ w−1 (f ) (mod xχ )(6.5)for all v, w and χ as above.
Using Lemma 6.2, we conclude that (Ω∗T (X))isomorphic to the subring of S WL consisting of those f such thatf ≡ sα (f ) (mod xα )Wis(6.6)for α ∈ Σ+ \ Σ+L and those f such thatf ≡ sα sθ(α) (f ) (mod xγ )(6.7)for γ = α − θ(α) ∈ ∆G/K . However, it follows from Lemma 6.3 that (6.6) holdsWfor all f ∈ S. We conclude from this that (Ω∗T (X)) is isomorphic to the subringWLof Sconsisting of those f such that (6.7) holds for γ = α − θ(α) ∈ ∆G/K .Doing the similar calculation for Y and using Lemma 6.2 and [23, Thm.
7.8]again, we see that (Ω∗T (Y ))WK is isomorphic to the same subring of S. Thiscompletes the proof of the theorem. Remark 6.5. Since Y is a smooth toric variety, Ω∗T (Y ) can be explicitly calculatedin terms of generators and relations using [25, Thm. 1.1]. Combining this withTheorem 6.4, one gets a simple way of computing the equivariant cobordism ringof wonderful symmetric varieties of minimal rank.Example 6.6. If G = PGL2 × PGL2 , and θ interchanges both factors, thenG/K ' PGL2 admits a unique wonderful compactification X = P3 . Namely, P3can be regarded as P(End(k 2 )), where G acts by left and right multiplications.EQUIVARIANT COBORDISM OF FLAG AND SYMMETRIC VARIETIES411The toric variety Y is P1 in this case.
The torus T ⊂ G is two-dimensional, andS = Lgr [[xα , xβ ]], where α and β are simple roots of G. Both Ω∗T (X) and Ω∗T (Y )can be computed explicitly:Ω∗T (X) ' Lgr [[x, xα , xβ ]]/((x − xα+β )(x − xα−β )(x − x−α+β )(x − x−α−β )),Ω∗T (Y ) ' Lgr [[x, xα , xβ ]]/((x − xα+β )(x − x−α−β )).The Weyl group W ' Z/2Z ⊕ Z/2Z changes signs of α and β. In particular, thenontrivial element of the Weyl group WK = diag(W ) acts on Ω∗T (Y ) by x 7→ x,xα 7→ x−α , xβ → x−β .
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J. 129 (2005), no. 2,249–290.Приложение G.Статья 7.Pavel Gusev, Valentina Kiritchenko, Vladlen Timorin “Counting verticesin the Gelfand-Zetlin polytopes”Journal of Combinatorial Theory, Series A 120 (2013) 960–969Разрешение на копирование: Согласноhttps://www.elsevier.com/about/policies/sharing автор статьи может использоватьполную журнальную версию статьи в своей диссертации при условии, чтоуказан DOI статьи.Journal of Combinatorial Theory, Series A 120 (2013) 960–969Contents lists available at SciVerse ScienceDirectJournal of Combinatorial Theory,Series Awww.elsevier.com/locate/jctaCounting vertices in Gelfand–Zetlin polytopesPavel Gusev a , Valentina Kiritchenko a,b , Vladlen Timorin a,caFaculty of Mathematics and Laboratory of Algebraic Geometry, National Research University Higher Schoolof Economics, 7 Vavilova St., 117312 Moscow, RussiabRAS Institute for Information Transmission Problems, Bolshoy Karetny Pereulok 19, 127994 Moscow, RussiacIndependent University of Moscow, Bolshoy Vlasyevskiy Pereulok 11, 119002 Moscow, Russiaa r t i c l ei n f oa b s t r a c tArticle history:Received 20 June 2012Available online 13 February 2013We discuss the problem of counting vertices in Gelfand–Zetlinpolytopes.
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