Showing papers by "Kirill Zainoulline published in 2012"
TL;DR: In this paper, the authors generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomologies, and algebraic cobordism.
Abstract: In the present paper we generalize the construction of the nil Hecke ring of Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The resulting object, which we call a formal (affine) Demazure algebra, is parameterized by a one-dimensional commutative formal group law and has the following important property: specialization to the additive and multiplicative periodic formal group laws yields completions of the nil Hecke and the 0-Hecke rings respectively. We also introduce a deformed version of the formal (affine) Demazure algebra, which we call a formal (affine) Hecke algebra. We show that the specialization of the formal (affine) Hecke algebra to the additive and multiplicative periodic formal group laws gives completions of the degenerate (affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We show that all formal affine Demazure algebras (and all formal affine Hecke algebras) become isomorphic over certain coefficient rings, proving an analogue of a result of Lusztig.
38 citations
TL;DR: In this paper, the notion of equivariant pretheory was introduced and studied in the context of rational cycles and G-torsors, and a ring which serves as an invariant of E was introduced.
Abstract: In the present paper we introduce and study the notion of an equivariant pretheory (basic examples are equivariant Chow groups of Edidin and Graham, Thomason’s equivariant K-theory and equivariant algebraic cobordism). Using the language of equivariant pretheories we generalize the theorem of Karpenko and Merkurjev on G-torsors and rational cycles. As an application, to every G-torsor E and a G-equivariant pretheory we associate a ring which serves as an invariant of E. In the case of Chow groups this ring encodes the information about the motivic J-invariant of E, in the case of Grothendieck’s K0 indexes of the respective Tits algebras and in the case of algebraic cobordism Ω it gives a quotient of the cobordism ring of G.
29 citations
TL;DR: In this article, the authors generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law.
Abstract: In the present paper we generalize the coproduct structure on nil Hecke rings introduced and studied by Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology theory and its associated formal group law. We then construct an algebraic model of the T-equivariant oriented cohomology of the variety of complete flags.
22 citations
TL;DR: In this paper, a connection between the indices of the Tits algebras of a semisimple linear algebraic group G and the degree one indices of its motivic J-invariant was established.
16 citations
Journal Article•
TL;DR: In this article, the relationship between i's and qi's over the integers is investigated and an application for the torsion of the Grothendieck -filtration and the Chow groups of some twisted flag varieties up to codimension 4.
Abstract: Consider a crystallographic root system together with its Weyl group W acting on the weight lattice �. Let Z(�) W and S(�) W be the W-invariant subrings of the integral group ring Z(�) and the symmetric algebra S(�) respectively. A celebrated result by Chevalley says that Z(�) W is a polynomial ring in classes of funda- mental representations �1;㨺㨻�n and S(�) W Q is a polynomial ring in basic polynomial invariants q1;㨺㨻qn. In the present paper we es- tablish and investigate the relationship betweeni's and qi's over the integers. As an application we provide estimates for the torsion of the Grothendieck -filtration and the Chow groups of some twisted flag varieties up to codimension 4.
10 citations
TL;DR: In this paper, the twisted γ -filtration on K 0 (G s ), where G s is a split simple linear algebraic group over a field k of characteristic prime to the order of the center of G s, was introduced and studied.
Abstract: In the present paper we introduce and study the twisted γ -filtration on K 0 ( G s ), where G s is a split simple linear algebraic group over a field k of characteristic prime to the order of the center of G s . We apply this filtration to construct nontrivial torsion elements in γ -rings of twisted flag varieties.
8 citations
Posted Content•
TL;DR: In this paper, the authors extended the notion of an exponent of the W-action introduced in [Baek-Neher-Zainoulline, arXiv:1106.4332] to the context of an algebraic oriented cohomology theory of a twisted flag variety.
Abstract: Let W be the Weyl group of a crystallographic root system acting on the associated weight lattice by reflections. In the present notes we extend the notion of an exponent of the W-action introduced in [Baek-Neher-Zainoulline, arXiv:1106.4332] to the context of an arbitrary algebraic oriented cohomology theory of Levine-Morel, Panin-Smirnov and the associated formal group law. From this point of view the classical Dynkin index of the associated Lie algebra will be the second exponent of the deformation map from the multiplicative to the additive formal group law. We apply this generalized exponent to study the torsion part of an arbitrary oriented cohomology theory of a twisted flag variety.
1 citations
Posted Content•
TL;DR: In the erratum, the authors correct a mistake due to a wrong choice of basic polynomial invariants over Z[1/2] in the original paper (v1).
Abstract: In the erratum we correct a mistake (due to a wrong choice of basic polynomial invariants over Z[1/2]) in the original paper (v1). Using the correct basic polynomial invariants we improve our results and bounds on the annihilator. We also simplify some of the proofs.
1 citations