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Orbit Sums and Modular Vector Invariants

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

Abstract

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
$$ \prod\limits_{j = 1}^n {\left( {1 + x_1 jz_1 + \cdots + x_{mj} z_m } \right) = 1 + \sum\limits_{1 \leqslant r_1 + \cdots + r_m \leqslant n} {u_{r1, \ldots rm} z_1^{r_1 } \cdots z_m^{r_m } .} } $$

Keywords

Polynomial invariants generalized orbit Chern classes finite groups Noether degree bound 

2000 Mathematics subject classification

11T55 13A50 14L24 16R30 20G40 

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Copyright information

© 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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