On the quiver Grassmannian in the acyclic case

Abstract

a b s t r a c tLet 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.

Degenerate flag varieties: moment graphs and Schröder numbers

“…[7,24,25]). Let Q be the equioriented type A n quiver with vertices labeled by numbers from 1 to n and arrows i → i + 1, i = 1, .…”

“…We conclude this section by describing the action of the torus T on the tangent space at a T -fixed point p S . Recall that the tangent space at p S is isomorphic to Hom(p S , M/p S ) where M = P ⊕ I ([9, Lemma 2.3], [7,25] (A, f, B) where A is a predecessor-closed connected sub-quiver of S, B is a successor-closed connected sub-quiver of θ M \ S and f : A → B is a quiver isomorphism compatible with π (see [11]). For example, in the left-hand side of the picture below…”

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“…The constructions and results in this section follow [Caldero and Reineke 2008;Schofield 1992]. Additionally to fix another dimension vector e such that e ≤ d componentwise, and define the Q 0 -graded Grassmannian Gr e (d) = i∈Q 0 Gr e i (M i ) which is a projective homogeneous space for G d of dimension i∈Q 0 e i (d i − e i ), namely Gr e (d) G d /P e for a maximal parabolic P e ⊂ G d .…”

“…We conclude this section by pointing out a useful isomorphism: let U be a point of Gr e (M) and let T U (Gr e (M)) denote the Zariski tangent space of Gr e (M) at U . As shown in [Schofield 1992;Caldero and Reineke 2008] we have the following scheme-theoretic description of the tangent space:…”

The Solution of the Quantum A 1 T-System for Arbitrary Boundary

“…Mutations are defined in such a way that the following Laurent property is guaranteed: any cluster data may be expressed as a Laurent polynomial of the cluster variables at any node of the tree. It was conjectured in [13] and proved in several particular cases (in particular in the so-called acyclic cases [3,15,1], or that of clusters arising from surfaces [22]) that these polynomials always have non-negative integer coefficients (Laurent positivity), a property that still awaits a good general combinatorial interpretation.…”