Partial flag varieties and preprojective algebras

Geiß, Christof; Leclerc, Bernard; Schröer, Jan

Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 825-876.

Soit $\Lambda$ une algèbre préprojective de type $A,D,E$, et soit $G$ le groupe algébrique complexe semi-simple et simplement connexe correspondant. Nous étudions les modules rigides des sous-catégories $\mathrm{Sub}Q$ où $Q$ désigne un $\Lambda$-module injectif, et nous introduisons une opération de mutation sur les modules rigides complets de $\mathrm{Sub}Q$. Ceci conduit à des structures d’algèbre amassée sur les anneaux de coordonnées des variétés de drapeaux partiels associées à $G$.

Let $\Lambda$ be a preprojective algebra of type $A,D,E$, and let $G$ be the corresponding semisimple simply connected complex algebraic group. We study rigid modules in subcategories $\mathrm{Sub}Q$ for $Q$ an injective $\Lambda$-module, and we introduce a mutation operation between complete rigid modules in $\mathrm{Sub}Q$. This yields cluster algebra structures on the coordinate rings of the partial flag varieties attached to $G$.

Classification : 14M15, 16D90, 16G20, 16G70, 17B10, 20G05, 20G20, 20G42
Keywords: Flag variety, preprojective algebra, Frobenius category, rigid module, mutation, cluster algebra, semicanonical basis
Mot clés : variété de drapeaux, algèbre préprojective, catégorie de Frobenius, module rigide, mutation, algèbre amassée, base semi-canonique

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