Ukrainian Mathematical Journal

, Volume 65, Issue 12, pp 1809–1821 | Cite as

On the Invariants of Root Subgroups of Finite Classical Groups

  • Jizhu Nan
  • Yufang Qin

We show that the invariant fields F q (X 1 , . . . ,X n ) G are purely transcendental over F q if G are root subgroups of finite classical groups. The key step is to find good similar groups of our groups. Moreover, the invariant rings of the root subgroups of special linear groups are shown to be polynomial rings and their corresponding Poincaré series are presented.


Normal Form Classical Group Unitary Group Galois Group Polynomial Ring 
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  1. 1.
    L. E. Dickson, “A fundamental system of invariants of the general modular linear group with a solution of the form problem,” Trans. Amer. Math. Soc., 12, 75–98 (1911).CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    G. C. Shephard and J. A. Todd, “Finite unitary reflection groups,” Can. J. Math., 6, 274–304 (1954).CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    L. G. Hua and Z. X. Wan, Classical Groups [in Chinese], Shanghai Sci. Technol. Press, Shanghai (1963).Google Scholar
  4. 4.
    H. Nakajima, “Invariants of finite groups generated by pseudo-reflections in positive characteristic,” Tsukuba J. Math., 3, 109–122 (1979).zbMATHMathSciNetGoogle Scholar
  5. 5.
    C. Wilkerson, “A primer on the Dickson invariants,” Amer. Math. Soc. Contemp. Math., 19, 421–434 (1983).CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    H. Chu, “Orthogonal group actions on rational function fields,” Bull. Inst. Math. Acad. Sinica, 16, 115–122 (1988).zbMATHMathSciNetGoogle Scholar
  7. 7.
    S. D. Cohen, “Rational function invariant under an orthogonal group,” Bull. London Math. Soc., 22, 217–221 (1990).CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D. Carlisle and P. H. Kropholler, “Rational invariants of certain orthogonal and unitary groups,” Bull. London Math. Soc., 24, 57–60 (1992).CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    G. Kemper, “Calculating invariant rings of finite groups over arbitrary fields,” J. Symb. Comput., 21, 351–366 (1996).CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    H. Chu, “Supplementary note on ‘rational invariants of certain orthogonal and unitary groups’,” Bull. London Math. Soc., 29, 37–42 (1997).CrossRefMathSciNetGoogle Scholar
  11. 11.
    S. Z. Li, Subgroup Structure of Classical Groups [in Chinese], Shanghai Sci. Technical Publ., Shanghai (1998).Google Scholar
  12. 12.
    S. M. Rajaei, “Rational invariants of certain orthogonal groups over finite fields of characteristic two,” Comm. Algebra, 28, 2367– 2393 (2000).CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    R. Kane, “Reflection groups and invariant theory,” CMS Books in Math., Springer-Verlag, New York (2001).Google Scholar
  14. 14.
    Z. M. Tang and Z. X. Wan, “A matrix approach to the rational invariants of certain classical groups over finite fields of characteristic two,” Finite Fields Their Appl., 12, 186–210 (2006).CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    J. Z. Nan and Y. Chen, “Rational invariants of certain classical similitude groups over finite fields,” Indiana Univ. Math. J., 4, 1947–1958 (2008).MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Jizhu Nan
    • 1
  • Yufang Qin
    • 1
  1. 1.Dalian University of TechnologyDalianChina

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