Showing papers by "Kirill Zainoulline published in 2011"
TL;DR: In this article, the twisted gamma-filtration was applied to construct torsion elements in the gamma-ring of an inner form of the Borel subgroups of a simple linear algebraic group.
Abstract: In the present notes we introduce and study the twisted gamma-filtration on K_0(G), where G is a split simple linear algebraic group over a field of characteristic prime to the order of the center of G. We apply this filtration to construct torsion elements in the gamma-ring of an inner form of the variety of Borel subgroups of G.
12 citations
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06 Apr 2011
TL;DR: In this article, a connection between the indices of the Tits algebras of a simple linear algebraic group G and the degree one parameters of its J-invariant was established.
Abstract: In the present paper we set up a connection between the indices of the Tits algebras of a simple linear algebraic group G and the degree one parameters of its J-invariant. Our main technical tool is the second Chern class map in the Riemann-Roch theorem without denominators. As an application we recover some known results on the J-invariant of quadratic forms of small dimension; we describe all possible values of the J-invariant of an algebra with involution up to degree 8 and give explicit examples; we establish several relations between the J-invariant of an algebra A with involution and the J-invariant (of the quadratic form) over the function field of the Severi-Brauer variety of A.
3 citations
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TL;DR: In this article, the authors established and investigated the relationship between polynomial invariants over rational numbers and Weyl groups over the integers, and provided an annihilator of the torsion part of the 3rd and 4th quotients of the Grothendieck gamma-filtration on the variety of Borel subgroups of the associated linear algebraic group.
Abstract: Consider a crystallographic root system together with its Weyl group $W$ acting on the weight lattice $M$. Let $Z[M]^W$ and $S^*(M)^W$ be the $W$-invariant subrings of the integral group ring $Z[M]$ and the symmetric algebra $S^*(M)$ respectively. A celebrated theorem of Chevalley says that $Z[M]^W$ is a polynomial ring over $Z$ in classes of fundamental representations $w_1,...,w_n$ and $S^*(M)^{W}$ over rational numbers is a polynomial ring in basic polynomial invariants $q_1,...,q_n$, where $n$ is the rank. In the present paper we establish and investigate the relationship between $w_i$'s and $q_i$'s over the integers. As an application we provide an annihilator of the torsion part of the 3rd and the 4th quotients of the Grothendieck gamma-filtration on the variety of Borel subgroups of the associated linear algebraic group.