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SL2-modules of small homological dimension

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Abstract

Let V n be the SL2-module of binary forms of degree n and let \( V = {V_{{n_1}}} \oplus \cdots \oplus {V_{{n_p}}} \). We consider the algebra \( R = \mathcal{O}{(V)^{{\text{S}}{{\text{L}}_2}}} \) of polynomial functions on V invariant under the action of SL2. The measure of the intricacy of these algebras is the length of their chains of syzygies, called homological dimension hd R. Popov gave in 1983 a classification of the cases in which hd R ≤ 10 for a single binary form (p = 1) or hd R ≤ 3 for a system of two or more binary forms (p > 1). We extend Popov’s result and determine for p = 1 the cases with hd R ≤ 100, and for p > 1 those with hd R ≤ 15. In these cases we give a set of homogeneous parameters and a set of generators for the algebra R.

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Authors and Affiliations

  1. Department of Mathematics, Eindhoven University of Technology, P. O. Box 513, 5600MB, Eindhoven, Netherlands

    Andries E. Brouwer

  2. Mathematisches Institut, Universität Basel, Rheinsprung 21, CH-4051, Basel, Switzerland

    Mihaela Popoviciu

Authors
  1. Andries E. Brouwer
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  2. Mihaela Popoviciu
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Correspondence to Andries E. Brouwer.

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To Professor T. A. Springer on the occasion of his 85th birthday

The second author is partially supported by the Swiss National Science Foundation.

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Brouwer, A.E., Popoviciu, M. SL2-modules of small homological dimension. Transformation Groups 16, 599–617 (2011). https://doi.org/10.1007/s00031-011-9138-5

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