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On relations of invariants for vector-valued forms. (English) Zbl 1044.15021

Let \(V\) and \(W\) be finite-dimensional vector spaces. Consider the space \(Bil(V,W)\) of bilinear maps from \(V\times V\) to \(W\). The group \(G=GL(V)\times GL(W)\) acts naturally in \(Bil(V,W)\). A relative invariant is a polynomial function \(f\) on \(Bil(V,W)\) such that \(f(gx)=\chi(g)f(x)\) for any \(g\in G\) and any \(x\in Bil(V,W)\), where \(\chi\) is a character of the group \(G\). Note that the linear span of all relative invariants is the algebra of invariants of the group \(G'=SL(V)\times SL(W)\).
T. Garrity and R. Mizner [Linear Algebra Appl. 218, 225–237 (1995; Zbl 0826.15023)] have given a description of a generating set for relative invariants. The present paper provides an algorithm for finding all relations between these generators.

MSC:

15A72 Vector and tensor algebra, theory of invariants
15A63 Quadratic and bilinear forms, inner products
14R20 Group actions on affine varieties
13A50 Actions of groups on commutative rings; invariant theory

Citations:

Zbl 0826.15023
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