Theory of invariants and applications

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We shall analyze the structure of the algebra of polynomial invariants of a classical group G such as GL_n, SO_n, O_n and Sp_{2n} acting on pairs of matrices (X,Y) of appropriate size (n or 2n) by simultaneous conjugation. In particular our intention is to compute (for some small values of n) the Poincare series, a minimal set of homogeneous generators, a homogeneous system of parameters, the corresponding Hironaka decomposition and also to describe the Hilbert's null-cone for the action. Such explicit results are known only in a handful of cases and it is important to have these data available for various applications. For instance these results may be used in the study of algebras satisfying polynomial identities.