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Abstract:
(Work joint with R. Basili)
We study the maximum Jordan partition Q(P) that can
occur for an element
A of the nilpotent commutator of an n by n nilpotent Jordan
matrix B of partition P.
Using results of R. Basili we show that Q(P)=P iff the parts of P differ by at least two.
We then define an auxiliary
integer matrix, Pow (P), related to the powers of A, and use it to determine the index (largest part) of Q(P).
The Hilbert function of K[A, B] is a natural invariant of the pair
(A, B). Using standard bases, we study
the partitions Pt associated to elements of the pencil A + t B, and we show
that Q(P) has decreasing parts.
P. Oblak has also recently, prior to our result, found the index
of Q(P).
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