Maximal green sequences, second Bruhat orders, and second Cambrian maps

Mikhail Gorsky (Amiens)

19-Apr-2022, 14:00-15:00 (23 months ago)

Abstract: This is a report on joint work in progress with Nicholas Williams.

Maximal 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. Demonet-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 Cambrian 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.

Higher versions of Bruhat and Cambrian orders in type A first appeared in late 80s in works of Manin-Shekhtman and Kapranov-Voevodsky, respectively. Kapranov and Voevodsky also defined a family of maps from higher Bruhat orders to higher Tamari-Stasheff orders (the latter are the higher versions of Cambrian orders in type A). More recently, second Bruhat orders showed up in works by Elias on (monoidal) categories of Soergel bimodules. I will explain how to realize a version of the second Bruhat order for a quiver Q as an order on equivalence classes of maximal green sequences for the corresponding preprojective algebra and the second Cambrian order as a similar order for 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 type A this provides, in a sense, an additive categorification of the corresponding Kapranov-Voevodsky map, which allows for a representation-theoretic interpretation of its fibers. If time permits, I will also explain how one can interpret second Cambrian orders in terms of polytopes and toric varieties associated with certain subword complexes.

combinatoricscategory theoryrepresentation theory

Audience: researchers in the topic

( slides )


The TRAC Seminar - Théorie de Représentations et ses Applications et Connections

Organizers: Thomas Brüstle*, Souheila Hassoun
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