Fomin, Sergey; Reading, Nathan Root systems and generalized associahedra. (English) Zbl 1147.52005 Miller, Ezra (ed.) et al., Geometric combinatorics. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Studies (ISBN 978-0-8218-3736-8/hbk). IAS/Park City Mathematics Series 13, 63-131 (2007). This article provides an excellent introduction to the theory of cluster algebras, initiated by S. Fomin and A. Zelevinsky [J. Am. Math. Soc. 15, No. 2, 497–529 (2002; Zbl 1021.16017)] in order to study total positivity and the dual canonical basis of a quantized enveloping algebra (in particular, its multiplicative structure). There is an emphasis on the combinatorial aspects of the theory and, in particular, the links with root systems of reflection groups and the generalized associahedra (or Stasheff polytopes) of the title. – The article is very accessible.For the entire collection see [Zbl 1126.05003]. Reviewer: Robert Marsh (Leeds) Cited in 59 Documents MSC: 52B11 \(n\)-dimensional polytopes 05E15 Combinatorial aspects of groups and algebras (MSC2010) 16G20 Representations of quivers and partially ordered sets 17B20 Simple, semisimple, reductive (super)algebras 17B37 Quantum groups (quantized enveloping algebras) and related deformations 20F55 Reflection and Coxeter groups (group-theoretic aspects) 52B12 Special polytopes (linear programming, centrally symmetric, etc.) Keywords:root systems; generalized associahedra; reflection groups; Coxeter groups; cluster algebras; convex polytopes; wiring diagrams; triangulations; double Bruhat cells; mutations; chamber minors Citations:Zbl 1021.16017 PDFBibTeX XMLCite \textit{S. Fomin} and \textit{N. Reading}, IAS/Park City Math. Ser. 13, 63--131 (2007; Zbl 1147.52005) Full Text: arXiv Online Encyclopedia of Integer Sequences: Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle. Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n). T(n,k) = binomial(n,k)*binomial(n+k,k), 0 <= k <= n, triangle read by rows. Triangle of binomial(n,k)*(binomial(n+k,k)-binomial(n+k-2,k-1)). Triangle of f-vectors of the simplicial complexes dual to the permutohedra of type B_n.