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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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