Séminaires : Séminaire d'Algèbre

Auslander and Reiten discovered that every Auslander–Gorenstein algebra admits a distinguished bijection between its indecomposable projective and injective modules, now known as the Auslander–Reiten bijection. In this talk, we present a new result showing that, for certain classes of algebras, the existence of such a bijection actually characterises the Auslander–Gorenstein property. We then relate this map to Iyama's grade bijection using a new homological characterisation of the Auslander-Gorenstein property. Afterwards, we discuss a new, linear algebraic interpretation of the Auslander–Gorenstein property and the Auslander–Reiten bijection using Coxeter matrices and their Bruhat decompositions. This approach opens the door to extending the definition of the Auslander–Reiten bijection to algebras that are not Auslander–Gorenstein.  This talk is mostly based on a joint work with R. Marczinzik and H. Thomas: https://arxiv.org/abs/2501.09447. 

This talk will take place in hybrid mode at the Institut Henri Poincaré.