Donaldson-Thomas theory and cluster algebras

By K. Nagao. 2010

127 citations recorded by the source corpus.

We provide a transformation formula of non-commutative Donaldson-Thomas invariants under a composition of mutations. Consequently, we get a description of a composition of cluster transformations in terms of quiver Grassmannians. As an application, we give an alternative proof of Fomin-Zelevinsky's conjectures on $F$-polynomials and $g$-vectors.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/800a0e9b4438c40882d5f691c53aad6fc9e97e57

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Citations in the corpus

Ordered Exchange Graphs (2013, background): “In parallel, instead of objects in C(Q,W ), Derksen, Weyman and Zelevinsky constructed in [28] the Caldero– Chapoton map for decorated representations over the Jacobian algebra J and gave the first complete proof of some of Fomin and Zelevinsky’s conjectures in the skewsymmetric case; Nagao construc”
Scattering diagrams, Hall algebras and stability conditions (2016, methodology): “Motivic Hall algebras of quiver representations were also used by Nagao [18].”
Lattice Properties of Oriented Exchange Graphs and Torsion Classes (2015, background): “an exchange graph is oriented according to the sign of the corresponding c-vector; see Section 2 for a more precise description. The fact that this orientation is well-dened is far from obvious (see [23]). The orientation of an exchange graph associated to a quiver is known to be acyclic. One approa”
Total positivity and cluster algebras (2010, background): “The Binet-Cauchy theorem implies that totally positive (resp., nonnegative) matrices in G = SLn form a multiplicative semigroup, denoted by G≥0.”
Cluster characters for cluster categories with infinite-dimensional morphism spaces (2010, result): “whose components are given by right multiplication by α. In this subsection, we prove the following theorem, which was ﬁrst formulated as a “hope” by K. Nagao in his message [31] and which is used in [32]. In a more restrictive setup, an analogous result was obtained in [23, Corollary 5.7]. CLUSTER ”
Positivity for quantum cluster algebras (2016, methodology): “′′ → 0 with M′ ∈ T and M′′ ∈ F. Let s = (s1,...,st) be a sequence of principal vertices of the ice quiver Q, and let W ∈ CQ/[CQ,CQ] be a nondegenerate algebraic potential with respect to s. Following [Nag13] we recursively deﬁne a torsion structure (Ts,Fs) on the category Jac(Q,W)-Mod, as well as a ”
Triangular bases in quantum cluster algebras and monoidal categorification conjectures (2015, background): “h that (1) T(t0) = T, (2) if two vertices t and t′are related by an edge labeled k, then the object T(t′) is obtained from T(t) by replacing one summand in a unique way. Theorem 2.2.2 ([DWZ10][Pla11b][Nag13]). The commutative cluster variables Xi(t)|q→1 agree with the cluster characters xT i(t). 2.3”
On tropical dualities in cluster algebras (2011, methodology): “cture (hence the property (1.8)) was proved in [2] for the case of skewsymmetric exchange matrices, using quivers with potentials and their representations (two diﬀerent proofs were recently given in [6, 8]). 4 TOMOKINAKANISHIANDANDREIZELEVINSKY We denote by CB;t0 t(resp. G B;t0 t ) the integer matr”
Exchange graphs and Ext quivers (2011, background): “ cluster exchange graph CEG(Q)∗ by Theorem 9.6, glue together to form EG◦(Γ 3 Q). Several of the underlying ideas in this section, including (9.1), already appear in the work of Plamondon [18], Nagao [17] and Keller [13]. 10. Constructionof A2-type exchangegraphvia the Farey graph In this section, w”
Cluster algebras and derived categories (2012, methodology): “goriﬁcation using decorated representations of quivers with potential, cf. below. Two diﬀerent proofs were given in [117] and, up to a technical extra hypothesis which is most probably superﬂuous, in [106]. In the case of valued quivers, the conjecture is open in general, but known to be true in man”
Universal geometric cluster algebras (2012, unknown): “Remark 8.14. Conjecture 8.8 (and thus Conjecture 8.1) is known in many cases, but currently not in full generality. In particular, it was proved in [4] for skew-symmetric exchange matrices. (See also [16, 18].) (In particular, since skew-symmetry is preserved under matrix mutation, Conjecture 8.11 i”
Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, result): “matrices. SILTING OBJECTS, SIMPLE-MINDED COLLECTIONS, t-STRUCTURES AND CO-t-STRUCTURES 5 Parallely, in the 3-Calabi–Yau setting, similar but more general results are obtained. Due to Amiot [5], Nagao [55] and Plamondon [58, 59], reachable silting objects in the fundamental domain are in one-to-one c”
Exact WKB analysis and cluster algebras (2014, background): “algebras 3 Hitchin system Schro¨dinger equation quadratic diﬀerentials triangulations of surfaces 3-CY categories cluster algebras [GMN13] [KT05] [KT05, GMN13, BS13] [LF12, BS13] [GSV05, FG06, FST08] [Nag10] [KS08] this work Figure 1. Outlines of previous and this works cluster algebras. See [Xie12,”