BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Mikhail Gorsky (Amiens) DTSTART:20220419T140000Z DTEND:20220419T150000Z DTSTAMP:20260423T005743Z UID:TRAC/42 DESCRIPTION:Title: Ma ximal green sequences\, second Bruhat orders\, and second Cambrian maps\nby Mikhail Gorsky (Amiens) as part of The TRAC Seminar - Théorie de Re présentations et ses Applications et Connections\n\n\nAbstract\nThis is a report on joint work in progress with Nicholas Williams.\n\nMaximal green sequences were introduced by Keller in the studies of connections between cluster algebras and quantum dilogarithm identities. In a broader sense\, such sequences are given by maximal chains in lattices of torsion classes tors A in module categories of finite-dimensional algebras A. Recently\, lattices of torsion classes have been a subject of intensive research. Dem onet-Iyama-Reading-Reiten-Thomas proved that for a quiver Q of Dynkin type ADE\, the lattice of torsion classes of its path algebra realizes the Cam brian lattice\, while the lattice tors Π for the preprojective algebra is isomorphic to the weak Bruhat order on the corresponding Weyl group. The Cambrian map\, introduced by Reading in combinatorial terms\, can thus be interpreted as a morphism of lattices of torsion classes.\n\nHigher versio ns of Bruhat and Cambrian orders in type A first appeared in late 80s in w orks of Manin-Shekhtman and Kapranov-Voevodsky\, respectively. Kapranov an d Voevodsky also defined a family of maps from higher Bruhat orders to hig her Tamari-Stasheff orders (the latter are the higher versions of Cambrian orders in type A). More recently\, second Bruhat orders showed up in work s by Elias on (monoidal) categories of Soergel bimodules. I will explain h ow to realize a version of the second Bruhat order for a quiver Q as an or der on equivalence classes of maximal green sequences for the correspondin g preprojective algebra and the second Cambrian order as a similar order f or the path algebra of Q. The second Cambrian map can then be interpreted as the "second level" of the Demonet-Iyama-Reading-Reiten-Thomas map. In t ype A this provides\, in a sense\, an additive categorification of the cor responding Kapranov-Voevodsky map\, which allows for a representation-theo retic interpretation of its fibers. If time permits\, I will also explain how one can interpret second Cambrian orders in terms of polytopes and tor ic varieties associated with certain subword complexes.\n LOCATION:https://researchseminars.org/talk/TRAC/42/ END:VEVENT END:VCALENDAR