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Defining Relations of Low Degree of Invariants of Two 4 times 4 Matrices
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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.
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Fermionic trace relations and supersymmetric indices at finite $N$
The supersymmetric index in a one-fermion matrix model for N=4 SYM is independent of N due to exact cancellations between bosonic and fermionic trace relations.
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