Counting using Hall algebras I. Quivers

Fei, Jiarui

doi:10.1016/j.jalgebra.2012.08.018

Cited by 8 publications

(11 citation statements)

References 12 publications

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“…[21]) that Gr γ (M ) is smooth. By the Poincaré duality and the quantum positivity [19,8], χ γ (M ) = 1 implies that Gr γ (M ) is zero-dimensional. By Lemma 4.2, we have that γ ⊥ β.…”

Section: Vertices Of N(m )mentioning

confidence: 99%

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Combinatorics of $F$-polynomials

Fei¹

2019

Preprint

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We introduce the stabilization functors to study the combinatorial aspect of the F -polynomial of a representation of any finite-dimensional basic algebra. We characterize the vertices of their Newton polytopes. We give an explicit formula for the F -polynomial restricting to any face of its Newton polytope. For acyclic quivers, we give a complete description of all facets of the Newton polytope when the representation is general. We also prove that the support of F -polynomial is saturated for any rigid representation. We provide many examples and counterexamples, and pose several conjectures.

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Section: Vertices Of N(m )mentioning

confidence: 99%

“…By the quantum positivity of Gr γ (M ) for rigid M [8,19], each Gr γ (M ) has positive Euler characteristic. Hence the support of F M is saturated as well.…”

Section: Saturation Definition 51 ([17]mentioning

confidence: 99%

Combinatorics of $F$-polynomials

Fei¹

2019

Preprint

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“…In [11], Ding and Xu showed that the comultiplication of H(A) implies the cluster multiplication formulas in A q (Q) (cf. also [13]). The expected connection has been established by Berenstein and Rupel [2].…”

Section: Introductionmentioning

confidence: 87%

“…The homomorphism and its generalization have been further investigated in [8,14,28]. Let us emphasize that the bialgebra structure of H(A) has also played a central role in [2,8,13,14].…”

Section: Introductionmentioning

confidence: 99%

Quantum cluster characters of Hall algebras revisited

Fu¹,

Peng²,

Zhang³

2020

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Let Q be a finite acyclic valued quiver. We give a bialgebra structure and an integration map on the Hall algebra associated to the morphism category of projective valued representations of Q. As an application, we recover the surjective homomorphism defined in [12], which realizes the principal coefficient quantum cluster algebra A q (Q) as a sub-quotient of the Hall algebra of morphisms. Moreover, we also recover the quantum Caldero-Chapoton formula, as well as their cluster multiplication formulas.

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“…In [5], we applied serval counting characters to the Harder-Narasimhan identity (2.1) in the Ringel-Hall algebra of a quiver and obtained several interesting formulas. All characters that we considered are originated from Reineke's counting character from the Hall algebra to certain quantum power series ring.…”

Section: Introductionmentioning

confidence: 99%

Counting using Hall Algebras II. Extensions from Quivers

Fei

2013

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We count the Fq-rational points of GIT quotients of quiver representations with relations. We focus on two types of algebras -one is one-point extended from a quiver Q, and the other is the Dynkin A 2 tensored with Q. For both, we obtain explicit formulas. We study when they are polynomial-count. We follow the similar line as in the first paper but algebraic manipulations in Ringel-Hall algebra will be replaced by corresponding geometric constructions.

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Counting using Hall algebras I. Quivers

Cited by 8 publications

References 12 publications

Combinatorics of $F$-polynomials

Combinatorics of $F$-polynomials

Quantum cluster characters of Hall algebras revisited

Counting using Hall Algebras II. Extensions from Quivers

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