Total positivity is by now a classical subject in linear algebra, having begun in earnest with the work of Gantmacher and Krein from 1937. Recent results of Postnikov and others have emphasised the importance of positivity in flag varieties, particularly the Grassmannian. A key tool in this area is Postnikov’s positroid stratification of the Grassmannian, and the cluster algebra structures on its various (open) cells, recently confirmed to exist by Galashin and Lam.
In this talk, I will explain this story in the language of representation theory, with the positroid varieties and their cluster algebra structures being encoded by the representation theory of various non-commutative orders over the power series ring in one variable. Except for the top-dimensional stratum, Galashin and Lam’s construction produces two different cluster algebra structures on each open positroid, and an application of this representation theoretic approach is a proof that these two structures quasi-coincide, as conjectured by Muller and Speyer in 2017. In particular, this means that these structures are equivalent from the point of view of total positivity.