On the Dimension of Coinvariants of Permutation Representations |
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Authors: | Larry Smith Michael Wibmer |
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Affiliation: | Universit?t G?ttingen, Germany Universit?t Innsbruck, Austria
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Abstract: | Let be a univariate, separable polynomial of degree n with roots x 1,…,x n in some algebraic closure of the ground field . 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 by permutation of the coordinates and each defines a relation P − P(x 1,…,x n ) 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 has dimension ‖G‖ over . 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 = Σ n acting via the tautological permutation representation. |
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Keywords: | 2000 Mathematics Subject Classification: 12F10 13A50 13P10 |
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