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

Garrity, Thomas; Grossman, Zachary

Electron. J. Linear Algebra 11, 24-40 (2004).

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.

Reviewer: Ivan Arzhantsev (Moskva)

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

Keywords:

vector-valued bilinear forms; relative invariants; first and second fundamental theorems of classical invariant theory

Citations:

Zbl 0826.15023