Computing with matrix invariants

Abstract

This is an improved version of the talk of the author given at the Antalya Algebra Days VII on May 21, 2005. We present an introduction to the theory of the invariants under the action of GL(n,C) by simultaneous conjugation of d matrices of size n x n. Then we survey some results, old or recent, obtained by a dozen of mathematicians, on minimal sets of generators, the defining relations of the algebras of invariants and on the multiplicities of the Hilbert series of these algebras. The picture is completely understood only in the case n=2. Besides, explicit minimal sets of generators are known for n=3 and any d and for n=4, d=2. The multiplicities of the Hilbert series are obtained only for n=3,4 and d=2. For n > 2 most of the concrete results are obtained with essential use of computers.