Grobner bases for the Hilbert ideal and coinvariants of the dihedral group D2p

Cataloged from PDF version of article.We consider a finite dimensional representation of the dihedral group D2p over a field of characteristic two where p is an odd integer and study the corresponding Hilbert ideal IH. We show that IH has a universal Gr¨obner basis consisting of invariants and monomials only. We provide sharp bounds for the degree of an element in this basis and in a minimal generating set for IH . We also compute the top degree of coinvariants when p is prime. c 2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Let V be a finite dimensional representation of a finite group G over a field F. There is an induced action of G on the symmetric algebra F[V ] of V ∗ that is given by g(f) = f ◦ g−1 for g ∈ G and f ∈ F[V ]. Let F[V ]G denote the ring of invariant polynomials in F[V ]. One of the main goals in invariant theory is to determine F[V ]G by computing the generators and relations. A closely related object is the Hilbert ideal, denoted IH , which is the ideal in F[V ] generated by invariants of positive degree. The Hilbert ideal often plays an important role in invariant theory as it is possible to extract information from it about the invariant ring. There is also substantial evidence that the Hilbert ideal is better behaved than the full invariant ring in terms of constructive complexity. The invariant ring is in general not generated by invariants of degree at most the group order when the characteristic of F divides the group order (this is known as the modular case) but it has been conjectured [2, Conjecture 3.8.6 (b)] that the Hilbert ideal always is. Apart from the non-modular case this conjecture is known to be true if V is a trivial source module [3] or if G = Zp and V is an indecomposable module [10]. Furthermore, Gr¨obner bases for IH have been determined for some classes of groups. The reduced Gr¨obner bases corresponding to several representations of Zp have been computed in a study of the module structure of the coinvariant ring F[V ]G which is defined to be F[V ]/IH , see [11]. The reduced Gr¨obner bases for the natural action of the symmetric and the alternating group can be found in [1] and [14], respectively. These bases have applications in coding theory, see [7]. In this paper we consider a representation of the dihedral group D2p over a field of characteristic two where p is an odd integer. Invariants of D2p in characteristic zero have been studied by Schmid [9] where she shows beyond other things that C[V ]D2 p is generated by invariants of degree at most p+1.More recently, bounds for the degrees of elements in both generating and separating sets over an algebraically closed field of characteristic two have been computed, see [6].We continue further in this direction and show that the Hilbert ideal IH is generated by invariants up to degree p and not less. We also construct a universal Gr¨obner basis for IH , i.e., a set G which forms a Gr¨obner basis of IH for any monomial order. Somewhat unexpectedly, the only polynomials that are not invariant in this set are monomials. Moreover, the maximal degree of a polynomial in this basis is p + 1. This is ∗ Corresponding author: e-mail: kohls@ma.tum.de, Phone: +49 89 289 17451, Fax: +49 89 289 17457 ∗∗ e-mail: sezer@fen.bilkent.edu.tr, Phone: +90 312 290 1085, Fax: +90 312 266 45 79