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On the Coxeter transformations for Tamari posets. (English) Zbl 1147.18007

The structure of the dendriform operad can be described by insertion of planar binary trees. The free dendriform algebra on one generator is an associative and also a Hopf algebra called the Hopf algebra of planar binary trees. The dendriform operad is an anticyclic operad which implies the existence of a linear map of order \(n+1\) on the vector space spanned by planar binary trees with \(n+1\) leaves.
The starting point of the paper under review is that the matrix of this endomorphism seems similar to a matrix appearing in the study of the Hopf algebra of planar binary trees. The main result gives that the linear maps obtained from the anticyclic structure of the dendriform operad can be described using Tamari posets. Such a poset consists of planar binary trees with a fixed number of leaves and partial order defined in terms of certain transformations on the trees. Considering Tamari posets as quivers with relations gives a family of Coxeter transformations on vector spaces spanned by planar binary trees. Up to a sign, iterating twice the Coxeter transformations recovers the anticyclic structure maps. As a consequence, the Coxeter transformation for Tamari posets is periodic.

MSC:

18D50 Operads (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
06A11 Algebraic aspects of posets
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16G20 Representations of quivers and partially ordered sets
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