P-flag spaces and incidence stratifications (Q2047271)

Let \(\mathbb F\) be a field and let \(P\) be a poset of cardinality \(n\). The authors define the notion of a \(P\)-flag in \(\mathbb F^n\), which is a collection of \(n\) subspaces in \(\mathbb F^n\) which satisfy suitable inclusions and dimensional conditions defined by \(P\). The space of \(P\)-flags has a homogeneous action of \(\mathrm{GL}(n,\mathbb F)\), which identifies it with \(\mathrm{GL}(n,\mathbb F)/I^*(P;\mathbb F)\) where \(I^*(P;\mathbb F)\) is the incidence group of \(P\). When \(P\) is a chain, the \(P\)-flags are the usual complete flags in \(\mathbb F^n\), and the incidence group is the group of upper triangular matrices. Then the authors define the incidence stratification associated to \(P\) on a Grassmannian \(\mathrm{Gr}_{\mathbb F}(k, n)\) and on a \(Q\)-flag space, where \(Q\) is another poset of cardinality \(n\). This general framework allows to deal with several combinatorial and geometric objects, unifying and extending different structures such as Bruhat orders, parking functions and weak orders on matroids.