BEGIN:VCALENDAR
VERSION:2.0
PRODID:researchseminars.org
CALSCALE:GREGORIAN
X-WR-CALNAME:researchseminars.org
BEGIN:VEVENT
SUMMARY:Bernhard Keller (Université de Paris)
DTSTART:20210223T160000Z
DTEND:20210223T170000Z
DTSTAMP:20260423T021138Z
UID:OCAS/19
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/OCAS/19/">Pr
 ogress on Leclerc's conjecture via Ménard's and Qin's theorems</a>\nby Be
 rnhard Keller (Université de Paris) as part of Online Cluster Algebra Sem
 inar (OCAS)\n\n\nAbstract\nIn 2014\, Leclerc conjectured the existence of 
 cluster structures for all open Richardson\nvarieties $R_{v\,w}$\, i.e. in
 tersections of a Schubert cell $C_w$ with an opposite\nSchubert cell $C^v$
  in a simple complex algebraic group which is simply connected and\nof sim
 ply laced type. Using representations of preprojective algebras\, he gave 
 a candidate\nseed for this structure and proved that the conjecture holds 
 when $v$ is less than or\nequal to $w$ in the weak right order. This holds
  in particular for open Schubert varieties\nin the Grassmannian. In this c
 ase\, Leclerc's seed was identified with a seed given by a\nplabic graph b
 y Serhiyenko--Sherman-Bennett--Williams (02/2019). This identification was
 \ngeneralized to open positroid varieties by Galashin--Lam (06/2019)\, who
  moreover proved\nLeclerc's conjecture for this class\, confirming a conje
 cture that had been known to\nthe experts since Scott's work (2006) and wa
 s put down in writing by Muller--Speyer (2017).\n\nIn his upcoming thesis\
 , using representations of preprojective algebras\,\nEtienne Ménard provi
 des an algorithm for the explicit computation of an initial seed\n(expecte
 d to agree with Leclerc's) in arbitrary type and shows that the correspond
 ing\nconjectural cluster structure is a cluster reduction of Geiss--Lecler
 c--Schröer's on the\nSchubert cell $C_w$. We will explain how this last r
 esult yields progress on Leclerc's conjecture\nfor Ménard's seed thanks t
 o Fan Qin's generic basis theorem and previous work by Muller\,\nPlamondon
 \, Geiss--Leclerc--Schröer\, Palu\, K--Reiten\, ... .\nThis is a report o
 n joint work with Peigen Cao.\n
LOCATION:https://researchseminars.org/talk/OCAS/19/
END:VEVENT
END:VCALENDAR