Donaldson-Thomas transformation of Grassmannian

Weng, Daping

doi:10.1016/j.aim.2021.107721

Cited by 4 publications

(5 citation statements)

References 33 publications

(34 reference statements)

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“…Following the work of Gross et al [GHKK18], if the Donaldson-Thomas transformation is a cluster transformation, then the Fock-Goncharov cluster duality conjecture holds. The cluster nature of Donaldson-Thomas transformations has been verified on many examples of cluster ensembles, including moduli spaces of G-local systems [GS18], Grassmannians [Wen21] and double Bruhat cells [Wen20]. As a direct consequence, the cluster duality conjecture holds in those cases.…”

Section: Donaldson-thomas Transformation and Periodicity Conjecturementioning

confidence: 94%

“…As explained in [Kel11, Section 5.7], Zamolodchikov's periodicity implies a result on the periodicity of the Donaldson-Thomas transformation. Weng [Wen21] gave a direct geometric proof of the periodicity of DT in the case of A A by realizing the Donaldson-Thomas transformation as a biregular automorphism on a configuration space of lines.…”

Section: Donaldson-thomas Transformation and Periodicity Conjecturementioning

confidence: 99%

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Cluster Structures on Double Bott–Samelson Cells

Shen

Weng

2021

Forum of Mathematics, Sigma

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Let $\mathsf {C}$ be a symmetrisable generalised Cartan matrix. We introduce four different versions of double Bott–Samelson cells for every pair of positive braids in the generalised braid group associated to $\mathsf {C}$ . We prove that the decorated double Bott–Samelson cells are smooth affine varieties, whose coordinate rings are naturally isomorphic to upper cluster algebras. We explicitly describe the Donaldson–Thomas transformations on double Bott–Samelson cells and prove that they are cluster transformations. As an application, we complete the proof of the Fock–Goncharov duality conjecture in these cases. We discover a periodicity phenomenon of the Donaldson–Thomas transformations on a family of double Bott–Samelson cells. We give a (rather simple) geometric proof of Zamolodchikov’s periodicity conjecture in the cases of $\Delta \square \mathrm {A}_r$ . When $\mathsf {C}$ is of type $\mathrm {A}$ , the double Bott–Samelson cells are isomorphic to Shende–Treumann–Zaslow’s moduli spaces of microlocal rank-1 constructible sheaves associated to Legendrian links. By counting their $\mathbb {F}_q$ -points we obtain rational functions that are Legendrian link invariants.

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Section: Donaldson-thomas Transformation and Periodicity Conjecturementioning

confidence: 94%

Section: Donaldson-thomas Transformation and Periodicity Conjecturementioning

confidence: 99%

Cluster Structures on Double Bott–Samelson Cells

Shen

Weng

2021

Forum of Mathematics, Sigma

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“…To this end, we are going to use the explicit formulas, in terms of cross-ratios, given by Volkov in [Vol1] for the Y-variables of the Y-system Y(A m , A n ). As in the A n -case, there are natural birational identifications between Conf m+n+2 (P m ) and the X-cluster variety of type A m ⊠ A n (see [GGSVV,§6.3] and [Wen,§2.3]): given a graph Γ of a certain type (namely, a 'minimal bipartite graph' ), one constructs a birational map x Γ : Conf m+n+2 (P m ) X A m ⊠A n . For instance, one can (and we will) consider the graph Γ 0 of [Wen,Example 2.11].…”

Section: Which Gives Us D Ymentioning

confidence: 99%

“…As in the A n -case, there are natural birational identifications between Conf m+n+2 (P m ) and the X-cluster variety of type A m ⊠ A n (see [GGSVV,§6.3] and [Wen,§2.3]): given a graph Γ of a certain type (namely, a 'minimal bipartite graph' ), one constructs a birational map x Γ : Conf m+n+2 (P m ) X A m ⊠A n . For instance, one can (and we will) consider the graph Γ 0 of [Wen,Example 2.11]. Composed with the natural projection…”

Section: Which Gives Us D Ymentioning

confidence: 99%

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On webs, polylogarithms and cluster algebras

Pirio

2021

Preprint

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In this text, we investigate webs which can be associated to cluster algebras from the point of view of the abelian functional equations these webs carry, focusing on the polylogarithmic ones. We introduce a general notion of webs whose rank is 'As Maximal as Possible' (AMP) and show that many webs associated either to polylogarithmic functional equations or to cluster algebras are of this type. In particular, we prove a few results and state some conjectures about cluster webs associated to (pairs of) Dynkin diagrams. Along the way, we show that many of the classical functional equations satisfied by low-order polylogarithms (such as Spence-Kummer's equation of the trilogarithm or the tetralogarithmic one of Kummer) are of cluster type.

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