Derived categories and symmetry: Grassmannians

We describe a straightforward algebraic procedure for constructing interesting symmetries of a particular derived category, specifically the derived category of coherent sheaves on the total space of a certain vector bundle over a Grassmannian. This can be thought of as a natural Сalabi–Yau category associated to the Grassmannian. We refer to our procedure as “window shifting”.
To explicitly describe the effect of a window shift we use some interesting complexes of locally free sheaves on the Grassmannian, variously known as staircase or Lascoux complexes. If there is time, we will also give a geometric description of the window shifts in terms of spherical twists, and explain how we hope to generalize our methods to a much wider class of examples. This is joint work with Ed Segal.