Computing with matrix invariants

TL;DR

This paper introduces the theory of invariants under the action of GL(n,C) on multiple matrices, surveys key results, and discusses the current understanding and computational approaches for different matrix sizes and numbers.

Contribution

It provides a comprehensive overview of known results on matrix invariants, highlighting explicit generators and Hilbert series multiplicities for specific cases, and emphasizes computational methods for larger cases.

Findings

Complete understanding for n=2

Explicit generators known for n=3,4 with any d and for n=4, d=2

Hilbert series multiplicities for n=3,4 and d=2

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.