Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866909599946244096 |
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| author | Li, Fang Pan, Jie |
| author_facet | Li, Fang Pan, Jie |
| contents | In this paper, we study the Newton polytopes of $F$-polynomials in a TSSS cluster algebra $\mathcal A$ and generalize them to a larger set consisting of polytopes $N_{h}$ associated to vectors $h\in\Z^{n}$ as well as $\widehat{\mathcal{P}}$ consisting of polytope functions $ρ_{h}$ corresponding to $N_{h}$.
The main contribution contains that
(i) obtaining a {\em recurrence construction} of the Laurent expression of a cluster variable in a cluster from its $g$-vector;
(ii) proving the subset $\mathcal{P}$ of $\widehat{\mathcal{P}}$ consisting of Laurent polynomials in $\widehat{\mathcal{P}}$ is a strongly positive $\Z Trop(Y)$-basis for $\mathcal{U}(\A)$ consisting of certain universally indecomposable Laurent polynomials when $\A$ is a cluster algebra with principal coefficients. For a cluster algebra $\mathcal A$ over arbitrary semifield $\mathbb P$ in general, $\mathcal{P}$ is a strongly positive $\Z¶$-basis for the intermediate cluster subalgebra $\mathcal{I_P(A)}$ of $\mathcal{U(A)}$. We call $\mathcal P$ the {\em polytope basis};
(iii) constructing some explicit maps among corresponding $F$-polynomials, $g$-vectors, $d$-vectors and cluster variables to characterize their relationship.
Moreover, we give three applications of (i), (ii) and (iii) respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2201_01440 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes Li, Fang Pan, Jie Representation Theory Commutative Algebra Combinatorics Rings and Algebras 13F60, 52B20 In this paper, we study the Newton polytopes of $F$-polynomials in a TSSS cluster algebra $\mathcal A$ and generalize them to a larger set consisting of polytopes $N_{h}$ associated to vectors $h\in\Z^{n}$ as well as $\widehat{\mathcal{P}}$ consisting of polytope functions $ρ_{h}$ corresponding to $N_{h}$. The main contribution contains that (i) obtaining a {\em recurrence construction} of the Laurent expression of a cluster variable in a cluster from its $g$-vector; (ii) proving the subset $\mathcal{P}$ of $\widehat{\mathcal{P}}$ consisting of Laurent polynomials in $\widehat{\mathcal{P}}$ is a strongly positive $\Z Trop(Y)$-basis for $\mathcal{U}(\A)$ consisting of certain universally indecomposable Laurent polynomials when $\A$ is a cluster algebra with principal coefficients. For a cluster algebra $\mathcal A$ over arbitrary semifield $\mathbb P$ in general, $\mathcal{P}$ is a strongly positive $\Z¶$-basis for the intermediate cluster subalgebra $\mathcal{I_P(A)}$ of $\mathcal{U(A)}$. We call $\mathcal P$ the {\em polytope basis}; (iii) constructing some explicit maps among corresponding $F$-polynomials, $g$-vectors, $d$-vectors and cluster variables to characterize their relationship. Moreover, we give three applications of (i), (ii) and (iii) respectively. |
| title | Recurrence formula, positivity and polytope basis in cluster algebras via Newton polytopes |
| topic | Representation Theory Commutative Algebra Combinatorics Rings and Algebras 13F60, 52B20 |
| url | https://arxiv.org/abs/2201.01440 |