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Gröbner bases for the rings of special orthogonal and \(2\times 2\) matrix invariants. (English) Zbl 1031.16016

Let the special orthogonal group \(\text{SO}_n\) act on \(k\)-tuples of vectors in \(n\)-dimensional space. The first fundamental theorem of invariant theory states that the ring of invariants is generated by the dot products of pairs of vectors and the determinants of all \(n\)-tuples of vectors. This paper is concerned with the ideal of relations between these generators. The generators of this ideal are given by the second fundamental theorem of algebra. However, in order to do explicit computations in this invariant ring, it is desirable to have a Gröbner basis of this ideal. The paper under review gives an explicit Gröbner basis. The cases \(n=3\) and \(n=4\) are interpreted as vector invariants for the \(2\times 2\) matrices. The authors also show that for \(n=2\), the invariant ring is a Koszul algebra by using SAGBI bases.

MSC:

16R30 Trace rings and invariant theory (associative rings and algebras)
16W22 Actions of groups and semigroups; invariant theory (associative rings and algebras)
13A50 Actions of groups on commutative rings; invariant theory
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References:

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