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Semicontinuous maps on module varieties
No full textWe study semicontinuous maps on varieties of modules over finite-dimensional algebras. We prove that truncated Euler maps are upper or lower semicontinuous. This implies that g-vectors and E-invariants of modules are upper semicontinuous. We also discuss inequalities of generic values of some upper semicontinuous maps
Quivers with potentials associated to triangulated surfaces, Part III: Tagged triangulations and cluster monomials
No full textTo 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)
Gentle Algebras Arising from Surfaces with Orbifold Points of Order 3, Part I: Scattering Diagrams
Get PDFCaldero-Chapoton algebras
Get PDFMotivated 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 cluster algebras
of Dynkin type, we associate a Caldero-Chapoton algebra CC(A) to any
(possibly infinite-dimensional) basic algebra A. By definition, CC(A) is (as a
vector space) generated by the Caldero-Chapoton functions CC(M) of the
decorated representations M of A. If A = P(Q,W) is the Jacobian algebra
defined by a 2-acyclic quiver Q with non-degenerate potential W, then we have
C(Q) ⊆ CC(A)⊆ U(Q) , where C(Q) and U(Q)
are the cluster algebra and the upper
cluster algebra associated to Q. The set B(A) 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 Schr ̈oer. 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 B(A) for arbitrary A, and we conjecture that B(A)
is a basis of the Caldero-Chapoton algebra CC(A). Thanks to the decomposition
theorem, all elements of B(A) can be seen as generalized cluster monomials. As
another application, we obtain a new proof for the sign-coherence of g-vectors
Linear independence of cluster monomials for skew-symmetric cluster algebras
No full textFomin-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 confrm these conjectures for all skew-symmetric cluster
algebras
Laminations of punctured surfaces as -reduced irreducible components
Get PDFLet be a surface with
marked points on the boundary,
and punctures , and an
arbitrary tagged triangulation of in the sense of
Fomin-Shapiro-Thurston. The Jacobian algebra
corresponding to the non-degenerate potential defined by Cerulli Irelli
and the second author is tame, as shown by Schr\"{o}er and the first two
authors. In this paper, we show that there is a natural isomorphism
of tame partial KRS-monoids that intertwines dual shear coordinates with
respect to , and generic -vectors of irreducible components. Here,
is the set of laminations of
considered by Musiker-Schiffler-Williams, with the
disjoint union of non-intersecting laminations as partial monoid operation. On
the other hand, denotes the set of
generically -reduced irreducible components of the decorated
representation varieties of , with the direct sum of generically
-orthogonal irreducible components as partial monoid operation, where is
the symmetrized -invariant of Derksen-Weyman-Zelevinsky,
.Comment: v2: Main result vastly generalized, from tagged triangulations of
signature zero, to arbitrary tagged triangulations; 42 pages, 13 figure
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