Central European Journal of Mathematics

, Volume 12, Issue 3, pp 421–428 | Cite as

Twisted gamma filtration and algebras with orthogonal involution

  • Caroline Junkins
Research Article


For the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in -rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.


Linear algebraic group Tits algebra Gamma filtration Grothendieck group Torsion Algebras with orthogonal involution 


20G15 14C25 14L30 16W10 


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  1. [1]
    Baek S., On the torsion of Chow groups of Severi-Brauer varieties, preprint available at
  2. [2]
    Baek S., Zainoulline K., Zhong C., On the torsion of Chow groups of twisted Spin-flags, preprint available at
  3. [3]
    Dejaiffe I., Somme orthogonale d’algèbres à involution et algèbre de Clifford, Comm. Algebra, 1998, 26(5), 1589–1612CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    Fulton W., Intersection Theory, 2nd ed., Ergeb. Math. Grenzgeb., 2, Springer, Berlin, 1998CrossRefzbMATHGoogle Scholar
  5. [5]
    Fulton W., Lang S., Riemann-Roch Algebra, Grundlehren Math. Wiss., 277,Springer, New York, 1985CrossRefzbMATHGoogle Scholar
  6. [6]
    Garibaldi S., Zainoulline K., The γ-filtration and the Rost invariant, J. Reine Angew. Math. (in press), DOI: 10.1515/crelle-2012-0114Google Scholar
  7. [7]
    Gille S., Zainoulline K., Equivariant pretheories and invariants of torsors, Transform. Groups, 2012, 17(2), 471–498CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    Grothendieck A., Théorie des Intersections et Théorème de Riemann-Roch, Lecture Notes in Math., 225, Springer, Berlin, 1971zbMATHGoogle Scholar
  9. [9]
    Junkins C., The J-invariant and Tits algebras for groups of inner type E6, Manuscripta Math., 2013, 140(1–2), 249–261CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    Karpenko N.A., Codimension 2 cycles on Severi-Brauer varieties, K-Theory, 1998, 13(4), 305–330CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    Karpenko N.A., Merkurjev A.S., Chow groups of projective quadrics, St. Petersburg Math. J., 1991, 2(3), 655–671MathSciNetGoogle Scholar
  12. [12]
    Knus M.-A., Merkurjev A., Rost M., Tignol J.-P., The Book of Involutions, Amer. Math. Soc. Colloq. Publ., 44, American Mathematical Society, Providence, 1998zbMATHGoogle Scholar
  13. [13]
    Panin I.A., On the algebraic K-theory of twisted flag varieties, K-Theory, 1994, 8(6), 541–585CrossRefzbMATHMathSciNetGoogle Scholar
  14. [14]
    Peyre E., Galois cohomology in degree three and homogeneous varieties, K-Theory, 1998, 15(2), 99–145CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    Quéguiner-Mathieu A., Semenov N., Zainoulline K., The J-invariant, Tits algebras and triality, J. Pure Appl. Algebra, 2012, 216(12), 2614–2628CrossRefzbMATHMathSciNetGoogle Scholar
  16. [16]
    Steinberg R., On a theorem of Pittie, Topology, 1975, 14(2), 173–177CrossRefzbMATHMathSciNetGoogle Scholar
  17. [17]
    Tits J., Représentations linéaires irréductibles d’un groupe réductif sur un corps quelconque, J. Reine Angew. Math., 1971, 247, 196–220zbMATHMathSciNetGoogle Scholar
  18. [18]
    Zainoulline K., Twisted gamma filtration of a linear algebraic group, Compos. Math., 2012, 148(5), 1645–1654CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Versita Warsaw and Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceUniversity of OttawaOttawaCanada

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