A Partial Order on Bipartitions From the Generalized Springer Correspondence

TL;DR

This paper establishes that the partial order on bipartitions derived from Lusztig's explicit formula for the generalized Springer correspondence coincides with the dominance order used in representation theory, clarifying their relationship.

Contribution

It proves the equivalence of two partial orders on bipartitions, connecting Lusztig's formula with the dominance order in a new, explicit way.

Findings

The partial order from Lusztig's formula matches the dominance order.

This equivalence clarifies the structure of unipotent classes in spin groups.

The result links geometric and combinatorial perspectives in representation theory.

Abstract

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.