Combinatorics of quasi-hereditary structures

Manuel Flores (1) , Yuta Kimura (1) , Baptiste Rognerud (2)

1 Universität Bielefeld = Bielefeld University
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

A quasi-hereditary algebra is an Artin algebra together with a partial order on its set of isomorphism classes of simple modules which satisfies certain conditions. In this article we investigate all the possible choices that yield quasi-hereditary structures on a given algebra, in particular we introduce and study what we call the poset of quasi-hereditary structures. Our techniques involve certain quiver decompositions and idempotent reductions. For a path algebra of Dynkin type A, we provide a full classification of its quasi-hereditary structures. For types D and E, we give a counting method for the number of quasi-hereditary structures. In the case of a hereditary incidence algebra, we present a necessary and sufficient condition for its poset of quasi-hereditary structures to be a lattice.

Mots clés: lattice; tilting module

Domaines: Combinatoire [math.CO]; Théorie des représentations [math.RT]

Manuel Flores, Yuta Kimura, Baptiste Rognerud. Combinatorics of quasi-hereditary structures. Journal of Combinatorial Theory, Series A, 2022, 187, pp.105559. ⟨10.1016/j.jcta.2021.105559⟩. ⟨hal-03481180⟩