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  "body": "\n## Definition\n\nThis is the coefficient-free, skew-symmetrizable version.\nLet $n \\ge 1$ and let $\\mathcal{F} = \\Q(u_1, \\dots, u_n)$ be the field of [[rational functions|0XDK9CGYKJ]] in $n$ indeterminates.\nAn integer $n \\times n$ [[matrix|0VD9ZSWWJG]] $B = (b_{ij})$ is *skew-symmetrizable* if there are positive integers $d_1, \\dots, d_n$ with $d_i b_{ij} = -d_j b_{ji}$ for all $i, j$.\nA *seed* is a pair $(\\mathbf{x}, B)$, where $\\mathbf{x} = (x_1, \\dots, x_n)$ is a free generating set of $\\mathcal{F}$ over $\\Q$ and $B$ is skew-symmetrizable.\nFor $k \\in \\set{1, \\dots, n}$, the *mutation* of $(\\mathbf{x}, B)$ at $k$ is the seed $(\\mathbf{x}', B')$ given as follows, where $[a]_+ = \\max(a, 0)$:\n\n1. $x'_i = x_i$ for $i \\ne k$, and $x'_k x_k = \\prod_{i} x_i^{[b_{ik}]_+} + \\prod_{i} x_i^{[-b_{ik}]_+}$.\n2. $b'_{ij} = -b_{ij}$ if $i = k$ or $j = k$.\n3. $b'_{ij} = b_{ij} + \\tfrac{1}{2}\\left(\\abs{b_{ik}} b_{kj} + b_{ik} \\abs{b_{kj}}\\right)$ otherwise.\n\nFix an initial seed $(\\mathbf{x}, B)$.\nThe *cluster variables* are the entries of all seeds reached from it by finite sequences of mutations, and each such $\\mathbf{x}'$ is a *cluster*.\nThe *cluster algebra* $\\mathcal{A}(B)$ is the subring of $\\mathcal{F}$ generated by all cluster variables.\n\n## Remarks\n\nExample: $n = 2$ and $B = \\begin{pmatrix} 0 \u0026 1 \\\\ -1 \u0026 0 \\end{pmatrix}$, which is skew-symmetric.\nMutation at $1$ gives $x'_1 = (1 + x_2)/x_1$, since $b_{21} = -1$ and $b_{11} = 0$.\nAlternating mutations give the cluster variables $x_1$, $x_2$, $(x_2 + 1)/x_1$, $(x_1 + x_2 + 1)/(x_1 x_2)$, $(x_1 + 1)/x_2$, five in all.\nThis is the cluster algebra of type $A_2$.\nNon-example: $B = \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ is not skew-symmetrizable, because $d_1 b_{12} = -d_2 b_{21}$ would give $d_1 = -d_2$ with $d_1, d_2 \u003e 0$.\nMutation is an involution: mutating twice at the same $k$ returns the original seed.\nBy the Laurent phenomenon of Fomin and Zelevinsky, every cluster variable is a Laurent polynomial with integer coefficients in the initial cluster.\nThe general definition also allows coefficients from a semifield.\n",
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