Irreducibility of Peterson–Richardson intersections

Determine the irreducibility behavior of Peterson–Richardson intersections in general, including which intersections are irreducible and which may have multiple irreducible components.

Background

The paper refines the Peterson-cell decomposition by considering Richardson-type intersections of Schubert and opposite Schubert cells, denoted by $$Y^0_{I,J}=(B^-w_JB/B)\cap(Bw_IB/B)\cap\mathrm{Pet}_n$$. These strata are central to the computation of Hilbert–Mumford weights and the description of semistable, stable, and polystable loci.

The authors note that irreducibility is not known for Peterson–Richardson intersections in general, and that some such intersections are in fact reducible. For the specific strata used in the paper—$Y^0_{I,\emptyset}$, $Y^0_{I,I}$, and the two endpoint strata $Y^0_{S,S\setminus\{\alpha_1\}}$ and $Y^0_{S,S\setminus\{\alpha_{n-1}\}}$—the required irreducibility follows directly. Thus, a broader determination of irreducibility remains unresolved beyond these cases.