On the Dimension of Coinvariants of Permutation Representations |
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Authors: | Larry Smith Michael Wibmer |
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Institution: | 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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