New elimination methods are applied to compute polynomial relations for the coefficients of the characteristic polynomial of certain families of matrices such as tensor squares.

The problems treated here concern polynomial relations between the coefficients of characteristic polynomials of certain naturally defined varieties of matrices such as tensor squares of matrices. These questions originated from the recognition problem of finite matrix groups, cf. [L-GO'B 97] and [PlR], where the general question is addressed. The reason why we take up the problem here with different examples is that these problems provide good challenges for testing elimination techniques. They arise in sequences parametrized by

The methods we apply are developed in [PlR]. In particular, elimination by ``degree steering'' is the preferred strategy here. Janet bases of the same ideal are computed repeatedly for different gradings of the polynomial ring. The degrees of the variables to be eliminated are increased in each step until a Janet basis for the elimination ideal can be read off. Other techniques which come up in the course of [PlR] still wait for implementation.

All results were obtained by using implementations of the involutive basis algorithm by V. P. Gerdt and Y. A. Blinkov [Ger 05], [GBY 01]. More precisely, all Janet bases have been computed by the new open source software package

- Tensor square of GL(
*n*,*K*) - Tensor square of SL(
*n*,*K*) - Compound representation of GL(
*n*,*K*) - Exterior square of SO(
*n*,*K*) - Reduced symmetric square of SO(
*n*,*K*)

[BCG 03] Y. A. Blinkov, C. F. Cid, V. P. Gerdt, W. Plesken, D. Robertz.

[ginv]

[Ger 05] V. P. Gerdt.

[GBY 01] V. P. Gerdt, Y. A. Blinkov, D. A. Yanovich.

[L-GO'B 97] C. R. Leedham-Green, E. O'Brien.

[PlR 05] W. Plesken, D. Robertz.

[PlR] W. Plesken, D. Robertz.