Daniel Labardini-Fragoso

Linear independence of cluster monomials for skew-symmetric cluster algebras

Fomin-Zelevinsky conjectured that in any cluster algebra, the cluster monomials are linearly independent and that the exchange graph and cluster complex are independent of the choice of coefficients. We confirm these conjectures for all skew-symmetric cluster algebras.

The representation type of Jacobian algebras

Abstract We show that the representation type of the Jacobian algebra P ( Q , S ) associated to a 2-acyclic quiver Q with non-degenerate potential S is invariant under QP-mutations. We prove that, apart from very few exceptions, P ( Q , S ) is of tame representation type if and only if Q is of finite mutation type. We also show that most quivers Q of finite mutation type admit only one non-degenerate potential up to weak right equivalence. In this case, the isomorphism class of P ( Q , S ) depends only on Q and not on S.

Quivers with potentials associated to triangulated surfaces, part IV: removing boundary assumptions

We prove that the quivers with potentials associated with triangulations of surfaces with marked points, and possibly empty boundary, are non-degenerate, provided the underlying surface with marked points is not a closed sphere with exactly five punctures. This is done by explicitly defining the QPs that correspond to tagged triangulations and proving that whenever two tagged triangulations are related to a flip, their associated QPs are related to the corresponding QP-mutation. As a by-product, for (arbitrarily punctured) surfaces with non-empty boundary, we obtain a proof of the non-degeneracy of the associated QPs which is independent from the one given by the author in the first paper of the series. The main tool used to prove the aforementioned compatibility between flips and QP-mutations is what we have called Popping Theorem, which, roughly speaking, says that an apparent lack of symmetry in the potentials arising from ideal triangulations with self-folded triangles can be fixed by a suitable right-equivalence.

Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials

To each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author’s previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin, Shapiro and Thurston (with an arbitrary system of coefficients).

Caldero-Chapoton algebras

Motivated by the representation theory of quivers with potential introduced by Derksen, Weyman and Zelevinsky and by work of Caldero and Chapoton, who gave explicit formulae for the cluster variables of Dynkin quivers, we associate a CalderoChapoton algebra AΛ to any (possibly infinite dimensional) basic algebra Λ. By definition, AΛ is (as a vector space) generated by the Caldero-Chapoton functions CΛ(M) of the decorated representations M of Λ. If Λ = P(Q,W ) is the Jacobian algebra defined by a 2-acyclic quiver Q with non-degenerate potential W , then we have AQ ⊆ AΛ ⊆ A Q , where AQ and A Q are the cluster algebra and the upper cluster algebra associated to Q. The set BΛ of generic Caldero-Chapoton functions is parametrized by the strongly reduced components of the varieties of representations of the Jacobian algebra P(Q,W ) and was introduced by Geiss, Leclerc and Schroer. Plamondon parametrized the strongly reduced components for finite-dimensional basic algebras. We generalize this to arbitrary basic algebras. Furthermore, we prove a decomposition theorem for strongly reduced components. We define BΛ for arbitrary Λ, and we conjecture that BΛ is a basis of the Caldero-Chapoton algebra AΛ. Thanks to the decomposition theorem, all elements of BΛ can be seen as generalized cluster monomials. As another application, we obtain a new proof for the sign-coherence of g-vectors. Caldero-Chapoton algebras lead to several general conjectures on cluster algebras.

CONES AND CONVEX BODIES WITH MODULAR FACE LATTICES.

If a convex body C has modular and irreducible face lattice (and is not strictly convex), there is a face-preserving homeomorphism from C to a section of a cone of hermitian matrices or C has dimension 8, 14 or 26.