Séminaires : Séminaire d'Algèbre

Given a plabic graph on R^2, we can choose a conormal lift of its zig-zag strands to the unit cotangent bundle of R^2, obtaining a Legendrian link. If the plabic graph satisfies a “grid” condition, its Legendrian link admits a natural embedding into the standard contact R^3. We study the Kashiwara-Schapira moduli space of microlocal rank 1 sheaves associated with the Legendrian link, and construct a natural (quasi-)cluster structure on this moduli space using Legendrian weaves. In particular, we prove that any braid variety associated with (beta Delta) for a 3-strand braid beta admits cluster structures with an explicit construction of initial seeds. We also construct Donaldson-Thomas transformations for these moduli spaces and prove that the upper cluster algebra equals its cluster algebra. In this talk, I will introduce the theoretical background and describe the basic combinatorics for constructing Legendrian weaves and the (quasi-)cluster structures from a grid plabic graph. This is based on joint work with Roger Casals, cf. arxiv.org/abs/2204.13244.