Defining Relations of Low Degree of Invariants of Two $4 \times 4$ Matrices

TL;DR

This paper investigates the algebra of invariants of two 4x4 matrices under conjugation, identifying all relations of degree less than 15 using advanced algorithms and computer algebra tools.

Contribution

It determines all relations of degree less than 15 in the invariant algebra of two 4x4 matrices, extending previous knowledge of the algebra's structure.

Findings

All relations of degree < 15 have been explicitly determined.

Developed algorithms based on representation theory for relation detection.

Used computer algebra to facilitate complex calculations.

Abstract

Over a field K of characteristic 0, we study the algebra of invariants of the general linear group GL(4,K) acting by simultaneous conjugation on two matrices of order 4. It coincides with the trace algebra generated by all traces of products of two generic matrices of order 4. It is known that the minimal degree of the defining relations of any homogeneous minimal generating set of this algebra is equal to 12. Starting with the generating set given recently by Drensky and Sadikova, we have determined all relations of degree < 15. For this purpose we have developed further algorithms based on representation theory of the general linear group and easy computer calculations with standard functions of Maple.