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.


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


Zbl 0826.15023
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