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Some elementary properties of Laurent phenomenon algebras
Rings and Algebras · Submitted 9 Jan 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We prove that each seed of $\mathcal{A}(\Sigma)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(\Sigma)$ with $n-1$ common cluster variables are connected with each other by one step of mutation.
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The abstract, sentence by sentence
Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(\Sigma)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(\Sigma)$ with $n-1$ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $\mathcal{U}(\Sigma)$ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $\Sigma$. Besides, we obtain the standard monomial bases of $\mathcal{L}(\Sigma)$. We also prove that $\mathcal{U}(\Sigma)$ coincides with $\mathcal{L}(\Sigma)$ under certain conditions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. | background 1.00 | 0.11 | 0.03 | asserted 0.76 |
| s1 | We prove that each seed of $\mathcal{A}(\Sigma)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(\Sigma)$ with $n-1$ common cluster variables are connected with each other by one step of mutation. | result 0.98 | 0.96 | 0.46 | asserted 1.00 |
| s2 | The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. | interpretation 0.43 | 0.68 | 0.05 | asserted 0.92 |
| s3 | Moreover, we show that $\mathcal{U}(\Sigma)$ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $\Sigma$. | result 0.98 | 0.93 | 0.12 | asserted 1.00 |
| s4 | Besides, we obtain the standard monomial bases of $\mathcal{L}(\Sigma)$. | result 0.92 | 0.96 | 0.09 | asserted 1.00 |
| s5 | We also prove that $\mathcal{U}(\Sigma)$ coincides with $\mathcal{L}(\Sigma)$ under certain conditions. | result 0.99 | 0.96 | 0.18 | asserted 1.00 |
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Original abstract
Let $\Sigma$ be Laurent phenomenon (LP) seed of rank $n$, $\mathcal{A}(\Sigma)$, $\mathcal{U}(\Sigma)$ and $\mathcal{L}(\Sigma)$ be its corresponding Laurent phenomenon algebra, upper bound and lower bound respectively. We prove that each seed of $\mathcal{A}(\Sigma)$ is uniquely defined by its cluster, and any two seeds of $\mathcal{A}(\Sigma)$ with $n-1$ common cluster variables are connected with each other by one step of mutation. The method in this paper also works for (totally sign-skew-symmetric) cluster algebras. Moreover, we show that $\mathcal{U}(\Sigma)$ is invariant under seed mutations when each exchange polynomials coincides with its exchange Laurent polynomials of $\Sigma$. Besides, we obtain the standard monomial bases of $\mathcal{L}(\Sigma)$. We also prove that $\mathcal{U}(\Sigma)$ coincides with $\mathcal{L}(\Sigma)$ under certain conditions.
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1 arXiv paper cite this in 1 sentence
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By the cluster formula (see Theorem 3.5 of [ref] ), we have $P_σ(B_t_1D^-1)P_σ^⊤=B_t_2D^-1$.
Some conjectures on generalized cluster algebras via the cluster formula and $D$-matrix pattern · On Theorem [ref] (1) · 0.83
Cites as background 3 papers
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The following notations, definitions and propositions can refer [ref] .
Laurent phenomenon algebras arising from surfaces · Laurent phenomenon algebra · 0.98
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We do not talk much about generalized cluster algebras here, and one can refer [ref] for details.
Teichm\"uller spaces of Riemann surfaces with orbifold points of arbitrary order and cluster variables · On Theorem [ref] (1) · 0.83
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We do not talk much about generalized cluster algebras here, and one can refer [ref] for details.
Some conjectures on generalized cluster algebras via the cluster formula and $D$-matrix pattern · On Theorem [ref] (1) · 0.83
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