Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Study on cluster algebras via abstract pattern and two conjectures on
  d-vectors and g-vector   We mainly introduce an abstract pattern to study cluster algebras. Cluster
algebras, generalized cluster algebras and Laurent phenomenon algebras are
unified in the language of generalized Laurent phenomenon algebras (briefly,
GLP algebras) from the perspective of Laurent phenomenon.
  In this general framework, we firstly prove that each positive and
d-vector-positive GLP algebra has the proper Laurent monomial property and thus
its cluster monomials are linearly independent. Skew-symmetric cluster algebras
are verified to be d-vector-positive, which gives the affirmation of Conjecture
[conjd] in [FZ3] in this case. And since the positivity of skew-symmetric
cluster algebras is well-known, the new proof is obtained for the linearly
independence of cluster monomials of skew-symmetric cluster algebras.
  For a class of GLP algebras which are pointed, g-vectors of cluster monomials
are defined. We verify that for any positive GLP algebra pointed at $t_0$, the
g-vectors ${\bf g}_{1;t}^{t_0},\cdots, {\bf g}_{n;t}^{t_0}$ form a $\mathbb
Z$-basis of $\mathbb Z^n$, and different cluster monomials have different
g-vectors. As a direct application, we verify that Conjecture [conjg] in [FZ3]
holds for skew-symmetrizable cluster algebras and acyclic sign-skew-symmetric
cluster algebras.

27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
17
The categories of this paper are: q-fin
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
21
The categories of this paper are: adap-org
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
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The categories of this paper are: chem-ph
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
16
The categories of this paper are: stat