Hongyi Zhao on Sun, 08 Jan 2023 05:45:47 +0100 |
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Determine the set of the conjugators between two affine unimodular matrix groups, whose linear parts are the same finite group belonging to the subgroup of GLNZ. |
Hi here, I have two affine unimodular matrix groups whose generators are as follows: gens2=[ [ -1, 0, 0, 0 ; 0, 1, 0, 1; 0, 0, 1, 0; 0, 0, 0, 1 ], [ -1, 0, 0, 0; 0, 1, 0, -1/2; 0, 0, -1, 0; 0, 0, 0, 1 ], [ -1, 0, 0, 0; 0, -1, 0, 1/2; 0, 0, 1, 0; 0, 0, 0, 1 ], [ 1, 0, 0, 1/2; 0, 1, 0, 1/2; 0, 0, 1, 1/2; 0, 0, 0, 1 ] ] gens3=[ [ 1, 0, 0, 0; -1, -1, 0, 0; 0, 0, 1, -1/2; 0, 0, 0, 1 ], [ -1, 0, 0, 0; 0, -1, 0, 1/2; 1, 0, 1, 0; 0, 0, 0, 1 ], [ 1, 0, 0, 0; -1, -1, 0, 0; -1, 0, -1, -1/2; 0, 0, 0, 1 ], [ 1, 0, 0, -1; 0, 1, 0, 1; 0, 0, 1, 1; 0, 0, 0, 1 ] ] # Sorry here once more, I really don't know what's corresponding syntax for this in PARI/GP: grp2=Group(gens2) grp3=Group(gens3) Here, the 3-by-3 matrices corresponding to linear parts of the above two set of generators form the same finite group G, which is a subgroup in GLNZ. I want to determine the set of the conjugators between these two groups, a.k.a., conjs = { c | grp2 ^ c = grp3 } Where, the linear part of c is an element of the normalizer of G, and the translation part of c is a rational vector which has the following form: [ x, y, z, 1 ] If there's no such conjugator, how can I make the decision quickly? Regards, Zhao -- Assoc. Prof. Hongsheng Zhao <hongyi.zhao@gmail.com> Theory and Simulation of Materials Hebei Vocational University of Technology and Engineering No. 473, Quannan West Street, Xindu District, Xingtai, Hebei province