Quiver Grassmannians and degenerate flag varieties.

*(English)*Zbl 1282.14083The work under review is a study of certain quiver Grassmannians. Let \(Q=\left( Q_{0},Q_{1}\right) \) be a quiver with \(n\) vertices and a finite set of arrows whose underlying graph is a Dynkin diagram. By identifying quiver Grassmannians with an \(\mathfrak{sl}_{n}\)-degenerate flag variety, the authors were led to consider the class of Grassmannians of subrepresentations of sums of projective and injective representations. This class has many desirable properties, as outlined in the results below.

Let \(P\) and \(I\) be a projective and an injective representation of \(Q\) respectively. It is proved that \(\text{Gr}_{\dim P}\left( P\oplus I\right) \) is a normal local complete intersection variety. Furthermore, its dimension is \(\sum_{i\in Q_{0}}d_{i}e_{i}-\sum_{\left( \alpha:i\rightarrow j\right) \in Q_{1}}d_{i}e_{j},\) where \(\dim P=\left( d_{i}\right) _{i\in Q_{0}},\;\dim I=\left( e_{i}\right) _{i\in Q_{0}}\) are the dimension vectors of \(P\) and \(I.\)

The Grassmannian \(\text{Gr}_{\dim P}\left( P\oplus I\right) \) can be stratified as follows. For \(N\in\text{Gr}_{\dim P}\left( P\oplus I\right) \) let \(N_{I}=N\cap I\) and \(N_{P}\) the image of the projection \(P\oplus I\rightarrow P.\) For a dimension vector \(f\), let \(\mathcal{S}_{f}\) be the elements \(N\in\text{Gr}_{\dim P}\left( P\oplus I\right) \) with \(\dim N_{I}=f,\;\dim N_{P}=\dim P-f.\) Then the natural map \(\mathcal{S} _{f}\rightarrow\text{Gr}_{f}\left( I\right) \times \text{Gr}_{P-f}\left( P\right) \) is shown to be a vector bundle, and the fibre over \(\left( N_{P},I_{P}\right) \) is isomorphic to \(\text{Hom}_{Q}\left( N_{P},I/N_{I}\right) .\) As the Poincaré polynomials of \(\text{Gr}_{f}\left( I\right) \) and \(\text{Gr}_{P-f}\left( P\right) \) can be easily computed, this provides an explicit computation of the Poincaré polynomial of \(\text{Gr}_{\dim P}\left( P\oplus I\right) .\)

Finally, the group Aut\(\left( P\oplus I\right) \) acts on \(\text{Gr}_{\dim P}\left( P\oplus I\right)\) algebraically. Let \(G\) be a specific subgroup of \(\text{Aut}\left( P\oplus I\right)\): one which is the whole automorphism group unless \(Q\) is of type \(A_{n}.\) It is shown that the action of \(G\) on \(\text{Gr}_{\dim P}\left( P\oplus I\right) \) has finitely many orbits. These orbits are parameterized by isomorphism classes \(\left( \left[ Q_{P}\right] ,\left[ N_{I}\right] \right) \) where \(Q_{P}\) is a quotient of \(P,\) \(N_{I}\) is a subrepresentation of \(I,\) and their dimensions coincide. Furthermore, if \(Q\) is equioriented of type \(A_{n}\), then the orbits are parametrized by torus fixed points.

Let \(P\) and \(I\) be a projective and an injective representation of \(Q\) respectively. It is proved that \(\text{Gr}_{\dim P}\left( P\oplus I\right) \) is a normal local complete intersection variety. Furthermore, its dimension is \(\sum_{i\in Q_{0}}d_{i}e_{i}-\sum_{\left( \alpha:i\rightarrow j\right) \in Q_{1}}d_{i}e_{j},\) where \(\dim P=\left( d_{i}\right) _{i\in Q_{0}},\;\dim I=\left( e_{i}\right) _{i\in Q_{0}}\) are the dimension vectors of \(P\) and \(I.\)

The Grassmannian \(\text{Gr}_{\dim P}\left( P\oplus I\right) \) can be stratified as follows. For \(N\in\text{Gr}_{\dim P}\left( P\oplus I\right) \) let \(N_{I}=N\cap I\) and \(N_{P}\) the image of the projection \(P\oplus I\rightarrow P.\) For a dimension vector \(f\), let \(\mathcal{S}_{f}\) be the elements \(N\in\text{Gr}_{\dim P}\left( P\oplus I\right) \) with \(\dim N_{I}=f,\;\dim N_{P}=\dim P-f.\) Then the natural map \(\mathcal{S} _{f}\rightarrow\text{Gr}_{f}\left( I\right) \times \text{Gr}_{P-f}\left( P\right) \) is shown to be a vector bundle, and the fibre over \(\left( N_{P},I_{P}\right) \) is isomorphic to \(\text{Hom}_{Q}\left( N_{P},I/N_{I}\right) .\) As the Poincaré polynomials of \(\text{Gr}_{f}\left( I\right) \) and \(\text{Gr}_{P-f}\left( P\right) \) can be easily computed, this provides an explicit computation of the Poincaré polynomial of \(\text{Gr}_{\dim P}\left( P\oplus I\right) .\)

Finally, the group Aut\(\left( P\oplus I\right) \) acts on \(\text{Gr}_{\dim P}\left( P\oplus I\right)\) algebraically. Let \(G\) be a specific subgroup of \(\text{Aut}\left( P\oplus I\right)\): one which is the whole automorphism group unless \(Q\) is of type \(A_{n}.\) It is shown that the action of \(G\) on \(\text{Gr}_{\dim P}\left( P\oplus I\right) \) has finitely many orbits. These orbits are parameterized by isomorphism classes \(\left( \left[ Q_{P}\right] ,\left[ N_{I}\right] \right) \) where \(Q_{P}\) is a quotient of \(P,\) \(N_{I}\) is a subrepresentation of \(I,\) and their dimensions coincide. Furthermore, if \(Q\) is equioriented of type \(A_{n}\), then the orbits are parametrized by torus fixed points.

Reviewer: Alan Koch (Decatur)