Poset Structure
Nilpotent orbits form a partially ordered set: given two nilpotent orbits, O1 is less than or equal to O2 if O1 is contained in the Zariski closure of O2. This poset has a unique minimal element, zero orbit, and unique maximal element, the regular nilpotent orbit, but in general, it is not a graded poset. If the ground field is algebraically closed then the zero orbit is covered by a unique orbit, called the minimal orbit, and the regular orbit covers a unique orbit, called the subregular orbit.
In the case of the special linear group SLn, the nilpotent orbits are parametrized by the partitions of n. By a theorem of Gerstenhaber, the ordering of the orbits corresponds to the dominance order on the partitions of n. Moreover, if G is an isometry group of a bilinear form, i.e. an orthogonal or symplectic subgroup of SLn, then its nilpotent orbits are parametrized by partitions of n satisfying a certain parity condition and the corresponding poset structure is induced by the dominance order on all partitions (this is a nontrivial theorem, due to Gerstenhaber and Hesselink).
Read more about this topic: Nilpotent Orbit
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