A Partial Order on Bipartitions From the Generalized Springer Correspondence

In \cite{Lusztig}, Lusztig gives an explicit formula for the bijection between the set of bipartitions and the set \(\mathcal{N}\) of unipotent classes in a spin group which carry irreducible local systems equivariant for the spin group but not equivariant for the special orthogonal group. The set \(\mathcal{N}\) has a natural partial order and therefore induces a partial order on bipartitions. We use the explicit formula given in \cite{Lusztig} to prove that this partial order on bipartitions is the same as the dominance order appeared in Dipper-James-Murphy's work.