The Betti table of a graded module over a polynomial ring encodes much of its structure and that of the corresponding sheaf on projective space.
In general, it is hard to tell which tables of numbers can arise as Betti tables. An easier problem is to describe such tables up to positive scalar multiple: the "cone of Betti tables". The Boij-Söderberg conjectures, proven by Eisenbud-Schreyer, gave a beautiful description of this cone and, as a bonus, a "dual" description of the cone of cohomology tables of sheaves.
I will describe recent extensions of this theory, joint with Nicolas Ford and Steven Sam, to the setting of GL-equivariant modules over coordinate rings of matrices. Here, the dual picture concerns sheaf cohomology on Grassmannians. The main result is an equivariant analog of the Boij-Söderberg pairing between Betti tables and cohomology tables. This is a bilinear pairing of cones, with output in the cone coming from the "base case" of square matrices, which we can fully characterize.