Orbit Sums and Modular Vector Invariants

  • Serguei A. Stepanov
Part of the Developments in Mathematics book series (DEVM, volume 16)


Let m, n positive integers, R a commutative ring with the unit element 1, and
$$ A_{mn} = R\left[ {x_{11} , \ldots ,x_{m1} ; \ldots ;x_{1n} , \ldots ,x_{mn} } \right] $$
the algebra of polynomials in mn variables xij over R. The symmetric group S n operates on the algebra A mn as a group of R-automorphisms by the rule: σ(xij) = xi,σ(j), σ G. Denote by \( A_{mn}^{S_n } \) the subalgebra of invariants of the algebra A mn with respect to S n and define polarized elementary symmetric polynomials \( u_{r_1 , \ldots ,r_m } \in A_{mn}^{S_n } \) in n vector variables (x11,..., xm1),..., (x1n,..., xmn) by means of the following formal identity
$$ \prod\limits_{j = 1}^n {\left( {1 + x_1 jz_1 + \cdots + x_{mj} z_m } \right) = 1 + \sum\limits_{1 \leqslant r_1 + \cdots + r_m \leqslant n} {u_{r1, \ldots rm} z_1^{r_1 } \cdots z_m^{r_m } .} } $$


Polynomial invariants generalized orbit Chern classes finite groups Noether degree bound 

2000 Mathematics subject classification

11T55 13A50 14L24 16R30 20G40 


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  1. 1.
    Campbell, H.E.A., Hughes, I., Pollack, R.D.: Vector invariants of symmetric groups. Can. Math. Bull. 33, 391–397 (1990)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Campbell, H.E.A., Hughes, I.P.: Vector invariants of U 2(F 2): a proof of a conjecture of Richman. Adv. Math. 126, 1–20 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Fleischmann, P.: A new degree bound for the vector invariants of symmetric groups. Trans. Am. Math. Soc. 350, 1703–1712 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fleischmann, P.: The Noether bound in invariant theory of finite groups. Adv. Math. 156, 23–32 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hilbert, D.: Über die vollen Invariantensysteme. Math. Ann. 42, 313–373 (1893)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Kemper, G.: Lower degree bounds for modular invariants and a question of I. Hughes. Transform. Groups 3, 135–144 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Noether, E.: Der Endlichkeitssatz der Invarianten endlicher Gruppen. Math. Ann. 77, 89–92 (1916)CrossRefGoogle Scholar
  8. 8.
    Noether, E.: Der Endlichkeitssatz der Invarianten endlicher linearer Gruppen der Characteristik p. Nachr. Ges. Wiss. Göttingen 1926, 28–35 (1926)Google Scholar
  9. 9.
    Richman, D.: On vector invariants over finite fields. Adv. Math. 81, 30–65 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Richman, D.: Invariants of finite groups over fields of characteristic p. Adv. Math. 124, 25–48 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Smith, L.: Polynomial Invariants of Finite Groups. A.K. Peters, Wellesley (1995)zbMATHGoogle Scholar
  12. 12.
    Stepanov, S.A.: Vector invariants of symmetric groups in prime characteristic. Discrete Math. Appl. 10, 455–468 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Weyl, H.: The Classical Groups, 2nd edn. Princeton University Press, Princeton (1953)Google Scholar

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© Springer-Verlag 2008

Authors and Affiliations

  • Serguei A. Stepanov
    • 1
    • 2
  1. 1.Department of MathematicsBilkent UniversityAnkaraTurkey
  2. 2.Department of AlgebraV. A. Steklov Mathematical Institute, Russian Academy of SciencesMoscowRussia

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