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$$Y$$-systems and generalized associahedra. (English) Zbl 1057.52003
A $$Y$$-system is a family $${\mathcal Y}$$ of commuting variables $$Y_i(t)$$, with $$i\in I$$ (a finite index set) and $$t\in\mathbb{Z}$$, which satisfy the recurrence relations $Y_i(t+1)Y_i(t-1)=\prod_{j\neq i}\bigl(Y_j(t)+ 1\bigr)^{-a_{ij}},$ where the $$a_{ij}$$ are the entries of an indecomposable Cartan matrix $$A$$ of finite type (and thus associated with an irreducible finite crystallographic Coxeter group $$G$$ and corresponding root system $$\Phi$$ of rank $$n$$, say). The main result of the paper verifies a conjecture of A. B. Zamolodchikov [Phys. Lett. B 253, No. 3–4, 391–394 (1991)] that this recurrence relation is periodic, with period $$h+2$$, where $$h$$ is the Coxeter number of the group $$G$$.
As shown by the authors, an equivalent result is the following. The Coxeter(-Dynkin) diagram of $$G$$ is a tree, and hence bipartite, and so the index set $$J$$ can partitioned as $$I=I_+\cup I_-$$. Writing $$\varepsilon(i)=\varepsilon$$ if $$i\in I_\varepsilon$$, defining the involution $\tau_\varepsilon(u_i): = \begin{cases}\frac{\prod_{j\neq i} (u_j+ 1)^{a_{ij}}} {u_i},\quad & \text{if } \varepsilon(i)=\varepsilon,\\ u_i,\quad & \text{otherwise},\end{cases}$ and letting $$w_0$$ denote the longest element in the Weyl group $$G$$ associated with $$A$$, then the period of $$\tau_-\tau_+$$ is $$(h+2)/2$$ if $$w_0=-1$$, or $$h+2$$ otherwise.
In the set $$\Phi$$ of roots, let $$\Pi$$ denote the set of simple roots and $$\Phi_{>0}$$ the set of positive roots. The authors construct a certain simplicial complex $$\Delta(\Phi)$$ with vertex-set $$\Phi_{\geq-1}: =\Phi_{>0}\cup (-\Pi)$$, derived from the $$Y$$-system $${\mathcal Y}$$. This complex is pure of dimension $$n-1$$; the maximal simplices (or clusters) are each $$\mathbb{Z}$$-bases of the root lattice, and form a complete simplicial fan in the ambient vector space. The authors conjecture that this fan is the normal fan of a simple $$n$$-polytope; this is the generalized associahedron. In a footnote, they remark that this conjecture has been established by F. Chapoton and the authors in Can. Math. Bull. 45, No. 4, 537–566 (2002; Zbl 1018.52007).

##### MSC:
 52B12 Special polytopes (linear programming, centrally symmetric, etc.) 20F55 Reflection and Coxeter groups (group-theoretic aspects)
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