Cluster structures on mixed invariant rings, and related combinatorics

Let V be a k-dimensional complex vector space. The Pluecker ring of polynomial SL(V) invariants of a collection of n vectors in V can be alternatively described as the homogeneous coordinate ring of the Grassmannian Gr(k,n). In 2003, using combinatorial tools developed by A. Postnikov, J. Scott showed that the Pluecker ring carries a cluster algebra structure. Over the ensuing decade, this has become one of the central examples of cluster algebra theory. In the 1930s, H. Weyl described the structure of the "mixed" Pluecker ring, the ring of polynomial SL(V) invariants of a collection of n vectors in V and m covectors in V*. We generalize Scott's construction and Postnikov's combinatorics to this more general setting. In particular, we show that each mixed Pluecker ring carries a natural cluster algebra structure. This was previously established by S. Fomin and P. Pylyavskyy in the case k=3.