Algebra Seminar: A derived equivalence for higher Auslander algebras of type A

Abstract: Given a positive integer d, a d-Auslander algebra A is characterized by the property that gldim A <= d+1 < = domdim A. Iyama showed that such an algebra is always obtained as the endomorphism algebra of a d-cluster tilting object in a module category. Starting from a linear A_n quiver, Iyama constructed an infinite family of d-Auslander algebras A_n^{d+1} called higher Auslander algebras of type A.

In this talk I will present a derived equivalence for the algebra A_{n+1}^d with gcd(n,d)=1. The derived equivalence is induced by a tilting complex T which is obtained by applying the Nakayama functor n+d times to a certain projective module P and taking the direct sum. We give a combinatorial model of A_{n+1}^d by lattice paths in a d by n rectangle. In this model P is determined by the rational Dyck paths. I will also give some ideas about the proof using this combinatorial model.

The seminar will be held via Zoom link (meeting ID: 645 5572 6999, please contact the organizer for passcode)

This is a seminar in our algebra seminar series.