Monatshefte für Mathematik

, Volume 151, Issue 1, pp 75–81 | Cite as

On the Dimension of Coinvariants of Permutation Representations

  • Larry Smith
  • Michael Wibmer
Article

Abstract.

Let \(f (Z) \in {\Bbb F}[Z]\) be a univariate, separable polynomial of degree n with roots x1,…,xn in some algebraic closure \(\bar{\Bbb F}\) of the ground field \({\Bbb F}\). It is a classical problem of Galois theory to find all the relations between the roots. It is known that the ideal of all such relations is generated by polynomials arising from G-invariant polynomials, where G is the Galois group of f(Z). Namely: The action of G on the ordered set of roots induces an action on \({\Bbb F}^n\) by permutation of the coordinates and each \(P \in {\Bbb F}[X_1, \ldots,X_n]^G\) defines a relation PP(x1,…,xn) called a G-invariant relation. These generate the ideal of all relations. In this note we show that the ideal of relations admits an H-basis of G-invariant relations if and only if the algebra of coinvariants \({\Bbb F}[X_1,\ldots,X_n]_G\) has dimension ‖G‖ over \({\Bbb F}\). To complete the picture we then show that the coinvariant algebra of a transitive permutation representation of a finite group G has dimension ‖G‖ if and only if G = Σn acting via the tautological permutation representation.

2000 Mathematics Subject Classification: 12F10, 13A50, 13P10 
Key words: Coinvariants, permutation representation, relation ideal 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • Larry Smith
    • 1
  • Michael Wibmer
    • 2
  1. 1.Universität GöttingenGermany
  2. 2.Universität InnsbruckAustria

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