Let $ \Sigma $ be a 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.
Citation: Qiuning Du, Fang Li. Some elementary properties of Laurent phenomenon algebras[J]. Electronic Research Archive, 2022, 30(8): 3019-3041. doi: 10.3934/era.2022153
Let $ \Sigma $ be a 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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Qiuning Du, Fang Li. Some elementary properties of Laurent phenomenon algebras[J]. Electronic Research Archive, 2022, 30(8): 3019-3041. doi: 10.3934/era.2022153
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