2013/08/22 by Myrto Kallipoliti, Kallipoliti, Myrto, Henri Mühle +1
Computer Science · Mathematics · #06A07 (Primary) 20F55 (Secondary) #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:06A07 #msc:20F55 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1308.4813
20 pages, 13 figures. The results of this paper are subsumed by arXiv:1312.2520, and it will therefore not be published
openalex publication_date 2013/08/22 · arxiv created 2014/08/12 · arxiv updated 2014/08/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For positive integers m and k, we introduce a family of lattices Ck(m) associated to the Cambrian lattice Ck of the dihedral group I2(k). We show that Ck(m) satisfies some basic properties of a Fuss-Catalan generalization of Ck, namely that Ck(1)=Ck and \lvertCk(m)|=Cat(m)(I2(k)). Subsequently, we prove some structural and topological properties of these lattices---namely that they are trim and EL-shellable---which were known for Ck before. Remarkably, our construction coincides in the case k=3 with the m-Tamari lattice of parameter 3 due to Bergeron and Préville-Ratelle. Eventually, we investigate this construction in the context of other Coxeter groups, in particular we conjecture that the lattice completion of the analogous construction for the symmetric group \mathfrakSn and the long cycle (1 2 … n) is isomorphic to the m-Tamari lattice of parameter n.