Graded cluster algebras

Grabowski, Jan E.

doi:10.48550/arxiv.1309.6170

Cited by 6 publications

(32 citation statements)

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“…Extend the grading additively to Laurent monomials hence to the cluster algebra A(x, y, B). In [6] it is proved that under this grading every exchange relation is homogeneous and thus the grading is compatible with mutation. The following two propositions come from the work in [15] to show that rank three non-acyclic cluster algebras have no maximal green sequences.…”

Section: Grading On Amentioning

confidence: 99%

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

Lee¹,

Li²,

Mills³

2015

SIGMA

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Section: Grading On Amentioning

confidence: 99%

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

Lee¹,

Li²,

Mills³

2015

SIGMA

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“…We recall briefly the concept of a Z r -grading on E cf. [21,22]. Choosing an initial seed ( B, {x i }) in E, such a choice of grading is determined by a r × (n + m) grading matrix G satisfying G B = 0.…”

Section: Normalized Seed Patternsmentioning

confidence: 99%

“…. , r (a standard grading is one that spans the kernel of the B-matrices, see [21]). We will not rely on gradings in what follows.…”

Section: Proofs For Sectionmentioning

confidence: 99%

Quasi-homomorphisms of cluster algebras

Fraser

2016

Advances in Applied Mathematics

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Abstract. We introduce quasi-homomorphisms of cluster algebras, a flexible notion of a map between cluster algebras of the same type (but with different coefficients). The definition is given in terms of seed orbits, the smallest equivalence classes of seeds on which the mutation rules for non-normalized seeds are unambiguous. We present examples of quasi-homomorphisms involving familiar cluster algebras, such as cluster structures on Grassmannians, and those associated with marked surfaces with boundary. We explore the related notion of a quasi-automorphism, and compare the resulting group with other groups of symmetries of cluster structures. For cluster algebras from surfaces, we determine the subgroup of quasi-automorphisms inside the tagged mapping class group of the surface.

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“…A tropical analogue of frieze patterns may be defined by replacing the bounding 1s by 0s and imposing the tropicalized relation d = max (b + c, 0) − a on each 2 × 2 subarray. Such tropical frieze patterns have attracted some interest lately (e.g., [Pro05,Guo13,AD13,Gra13]).…”

Section: Tropical Frieze Patternsmentioning

confidence: 99%

Cyclic sieving of increasing tableaux and small Schröder paths

Pechenik

2014

Journal of Combinatorial Theory, Series A

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Abstract. An increasing tableau is a semistandard tableau with strictly increasing rows and columns. It is well known that the Catalan numbers enumerate both rectangular standard Young tableaux of two rows and also Dyck paths. We generalize this to a bijection between rectangular 2-row increasing tableaux and small Schröder paths. We demonstrate relations between the jeu de taquin for increasing tableaux developed by H. Thomas and A. Yong and the combinatorics of tropical frieze patterns. We then use this jeu de taquin to present new instances of the cyclic sieving phenomenon of V. Reiner, D. Stanton, and D. White, generalizing results of D. White and of J. Stembridge.

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Graded cluster algebras

Cited by 6 publications

References 0 publications

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

A Combinatorial Formula for Certain Elements of Upper Cluster Algebras

Quasi-homomorphisms of cluster algebras

Cyclic sieving of increasing tableaux and small Schröder paths

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