Properties of The Dominance Ordering
- Among the partitions of n, (1,…,1) is the smallest and (n) is the largest.
- The dominance ordering implies lexicographical ordering, i.e. if p dominates q and p ≠ q, then for the smallest i such that pi ≠ qi one has pi > qi.
- The poset of partitions of n is linearly ordered (and is equivalent to lexicographical ordering) if and only if n ≤ 5. It is graded if and only if n ≤ 6. See image at right for an example.
- A partition p covers a partition q if and only if pi = qi + 1, pk = qk − 1, pj = qj for all j ≠ i,k and either (1) k = i + 1 or (2) qi = qk (Brylawski, Prop. 2.3). Starting from the Young diagram of q, the Young diagram of p is obtained from it by first removing the last box of row k and then appending it either to the end of the immediately preceding row k − 1, or to the end of row i < k if the rows i through k of the Young diagram of q all have the same length.
- Every partition p has a conjugate (or dual) partition p′, whose Young diagram is the transpose of the Young diagram of p. This operation reverses the dominance ordering:
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- if and only if
- The dominance ordering determines the inclusions between the Zariski closures of the conjugacy classes of nilpotent matrices.
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