## 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

Zbl 0826.15023
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