On the quiver Grassmannian in the acyclic case

2006/11/03 by Philippe Caldero, Markus Reineke, Caldero, Philippe +1 · 3 citations
Mathematics · #14L30 #16G20 #16G70 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.math/0611074

openalex publication_date 2006/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.

Citations

• Laurent expansions in cluster algebras via quiver representations

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• PrIncipal quiver Grassmannians: conjectures
• From brick manifolds to Grassmannians of bimodules
• Regularity in Linear Degenerations of Flag Variety

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