[GAP Forum] Constructing nilpotent endomorphisms given a partition in GAP

[Neil Saunders]

Dear GAP Forum,

My apologies if these are easy questions, but I’m very new to computing with GAP.

I would like to construct a nilpotent endomorphism N acting on an even dimensional vector space V, that is self-adjoint with respect to a symplectic form.

Up to conjugacy, the Jordan blocks of a nilpotent matrix N are given by a partition \lambda of some non-negative integer.  So I would like to be able to construct the nilpotent endomorphism as follows:

  *   Given a partition \lamba, construct nilpotent matrix N whose Jordan blocks have size the parts of the doubled partition \lambda \cup \lambda (doubled partition because the vector space is even dimensional);
  *   A bilinear form on the (preferably rational) vector space such that N is self-adjoint with respect to this bilinear form  (I understand that while GAP can construct vector spaces over the rationals, it only constructs bilinear forms on vector spaces over finite fields?) I was trying to use the BilinearFormByMatrix function, but was finding it hard to control the basis with respect to the matrix N in that case.

Any help/tips with GAP’s inbuilt functions would be greatly appreciated. Again, apologies if these are naive questions but I’ve not done any serious computation in GAP before and am doing my best to learn the ropes.

Best wishes,
Neil

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