Progress on Leclerc's conjecture via Ménard's and Qin's theorems

Abstract: In 2014, Leclerc conjectured the existence of cluster structures for all open Richardson varieties $R_{v,w}$, i.e. intersections of a Schubert cell $C_w$ with an opposite Schubert cell $C^v$ in a simple complex algebraic group which is simply connected and of simply laced type. Using representations of preprojective algebras, he gave a candidate seed for this structure and proved that the conjecture holds when $v$ is less than or equal to $w$ in the weak right order. This holds in particular for open Schubert varieties in the Grassmannian. In this case, Leclerc's seed was identified with a seed given by a plabic graph by Serhiyenko--Sherman-Bennett--Williams (02/2019). This identification was generalized to open positroid varieties by Galashin--Lam (06/2019), who moreover proved Leclerc's conjecture for this class, confirming a conjecture that had been known to the experts since Scott's work (2006) and was put down in writing by Muller--Speyer (2017).

In his upcoming thesis, using representations of preprojective algebras, Etienne Ménard provides an algorithm for the explicit computation of an initial seed (expected to agree with Leclerc's) in arbitrary type and shows that the corresponding conjectural cluster structure is a cluster reduction of Geiss--Leclerc--Schröer's on the Schubert cell $C_w$. We will explain how this last result yields progress on Leclerc's conjecture for Ménard's seed thanks to Fan Qin's generic basis theorem and previous work by Muller, Plamondon, Geiss--Leclerc--Schröer, Palu, K--Reiten, ... . This is a report on joint work with Peigen Cao.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)