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数学代写|几何组合代写Geometric Combinatorics代考|Cluster Complexes and Generalized Associahedra

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数学代写|几何组合代写Geometric Combinatorics代考|Cluster Complexes and Generalized Associahedra

This section is based on $[\mathbf{1 9}, \mathbf{2 1}]$, except for the last statement in Theorem 4.11, which was proved in $[\mathbf{1 3}]$.

It can be shown that in a given cluster algebra of finite type, each seed is uniquely determined by its cluster. Consequently, the combinatorics of exchanges is encoded by the cluster complex, a simplicial complex (indeed, a pseudomanifold) on the set of all cluster variables whose maximal simplices (facets) are the clusters. See Figure 4.3. By Theorem 4.10, the cluster variables-hence the vertices of the cluster complex-can be naturally labeled by the set $\Phi_{\geq-1}$ of “almost positive roots” in the associated root system $\Phi$.

This dual graph of the cluster complex is precisely the exchange graph of the cluster algebra.

Theorem 4.11 below shows that the cluster complex is always spherical, and moreover polytopal.

Recall that $Q_{\mathbb{R}}$ denotes the $\mathbb{R}$-span of $\Phi$. The $\mathbb{Z}$-span of $\Phi$ is the root lattice, denoted by $Q$.

数学代写|几何组合代写Geometric Combinatorics代考|Polytopal Realizations of Generalized Associahedra

We now demonstrate how to explicitly describe each generalized associahedron by a set of linear inequalities.

Theorem 4.17. Suppose that a $\left(-w_{\circ}\right)$-invariant function $F:-\Pi \rightarrow \mathbb{R}$ satisfies the inequalities
$$\sum_{i \in I} a_{i j} F\left(-\alpha_i\right)>0 \quad \text { for all } j \in I .$$
Let us extend $F$ (uniquely) to a $\left\langle\tau_{-}, \tau_{+}\right\rangle$-invariant function on $\Phi_{\geq-1}$. The generalized associahedron is then given in the dual space $Q_{\mathbb{R}}^*$ by the linear inequalities
$$\langle\mathbf{z}, \alpha\rangle \leq F(\alpha), \text { for all } \alpha \in \Phi_{\geq-1}$$
An example of a function $F$ satisfying the conditions in Theorem 4.17 is obtained by setting $F\left(-\alpha_i\right)$ equal to the coefficient of the simple coroot $\alpha_i^{\vee}$ in the half-sum of all positive coroots. (Coroots are the roots of the “dual” root system; see $[\mathbf{9}, \mathbf{3 4}]$.

Example 4.18. In type $A_3$, Theorem 4.17 is illustrated in Figure 4.7, which shows a 3-dimensional associahedron given by the inequalities
\begin{aligned} \max \left(-z_1,-z_3, z_1, z_3, z_1+z_2, z_2+z_3\right) & \leq 3 / 2 \ \max \left(-z_2, z_2, z_1+z_2+z_3\right) & \leq 2 . \end{aligned}

数学代写|几何组合代写Geometric Combinatorics代考|Polytopal Realizations of Generalized Associahedra

$$\sum_{i \in I} a_{i j} F\left(-\alpha_i\right)>0 \quad \text { for all } j \in I .$$

$$\langle\mathbf{z}, \alpha\rangle \leq F(\alpha), \text { for all } \alpha \in \Phi_{\geq-1}$$

$$\max \left(-z_1,-z_3, z_1, z_3, z_1+z_2, z_2+z_3\right) \leq 3 / 2 \max \left(-z_2, z_2, z_1+z_2+z_3\right) \quad \leq 2$$

MATLAB代写

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