BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Nicholas Williams (Lancaster University) DTSTART:20230829T073000Z DTEND:20230829T083000Z DTSTAMP:20260423T021304Z UID:OISTRTS/48 DESCRIPTION:Title: Higher-dimensional cluster combinatorics and representation theory\nb y Nicholas Williams (Lancaster University) as part of OIST representation theory seminar\n\n\nAbstract\nPerhaps the most prominent example of a clus ter algebra is the type A cluster algebra\, where clusters are in bijectio n with triangulations of a convex polygon\, as observed by Fomin and Zelev insky. A categorical version of this relationship is that triangulations o f a convex polygon are in bijection with cluster-tilting objects in the cl uster category of the path algebra of the type A quiver. In each case\, mu tating the cluster or cluster-tilting object corresponds to flipping a dia gonal inside a quadrilateral. It is natural to wonder whether any similar relationship exists for triangulations of higher-dimensional polytopes. In deed\, in a beautiful paper Oppermann and Thomas show that triangulations of even-dimensional cyclic polytopes are in bijection with cluster-tilting objects in the cluster categories of the higher Auslander algebras of typ e A\, which were introduced by Iyama. Mutating the cluster-tilting objects corresponds to bistellar flips of triangulations\, which are the higher-d imensional analogues of flipping a diagonal inside a quadrilateral. In thi s talk\, we will outline the work of Oppermann and Thomas\, and explain th e odd-dimensional half of the picture too. Indeed\, the speaker has shown that triangulations of odd-dimensional cyclic polytopes are in bijection w ith equivalence classes of maximal green sequences for the higher Auslande r algebras of type A\, where maximal green sequences are maximal chains of cluster-tilting objects.\n LOCATION:https://researchseminars.org/talk/OISTRTS/48/ END:VEVENT END:VCALENDAR