Thin Schubert cells of codimension two (Q1012024)

Denote by \([n]\) the set of integers \(\{1,\dots,n\}\) and fix a base \(\{e_1, \dots,e_n\}\) of the complex vector space \(C^n\). For each subset \(I \subset [n]\) consider the vector space \(E_I\) spanned by the set \(\{e_i\}_{i\in I}\). With this notation one can define \textit{thin Schubert cells} in the following way: two \(k\)-dimensional vector subspaces \(L_1, L_2 \subset C^n\) lie in the same thin Schubert cell if \(\dim(L_1 \cap E_I)=\dim(L_2 \cap E_I)\) for all \(I \subset [n]\). These thin Schubert cells provide a decomposition of the Grassmannian \(G(k,n)\) of \(k\)-vector spaces of \(C^n\) finer than the usual given by Schubert cells. Given a thin Schubert cell \({\mathcal L}\), a set of integers \(d({\mathcal L})=\{d_I\}_{I \subset [n]}\) is produced, being \(d_I=\dim(L \cap E_I)\) for any \(L \in {\mathcal L}\). This set \(d({\mathcal L})\) satisfies what is called \textit{matroid conditions} (see Def. 2.1 for details). Conversely, given a matroid \(\underline{d}=\{d_I\}_{I \subset[n]}\), it is possible to construct \({\mathcal L}(\underline d)=\{L \in G(k,n):\dim(L \cap E_I)=d_I\) for all \(I\subset [n]\}\) which is a thin Schubert cell when non empty. The problem of finding necessary and sufficient condition for \({\mathcal L}(\underline d)\) to be nonempty is the main goal of this paper. In Thm. 3.1 this is solved for \(k=n-2\). Section 4 is devoted to study of the question of the decomposition in thin Schubert cells of the closure of a thin Schubert cell.