27 research outputs found

    The Discrete Fundamental Group of the Associahedron, and the Exchange Module

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    The associahedron is an object that has been well studied and has numerous applications, particularly in the theory of operads, the study of non-crossing partitions, lattice theory and more recently in the study of cluster algebras. We approach the associahedron from the point of view of discrete homotopy theory. We study the abelianization of the discrete fundamental group, and show that it is free abelian of rank (n+24)\binom{n+2}{4}. We also find a combinatorial description for a basis of this rank. We also introduce the exchange module of the type AnA_n cluster algebra, used to model the relations in the cluster algebra. We use the discrete fundamental group to the study of exchange module, and show that it is also free abelian of rank (n+23)\binom{n+2}{3}.Comment: 16 pages, 4 figure

    Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups

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    This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset NC(k)(W)NC^{(k)}(W) for each finite Coxeter group WW and for each positive integer kk. When k=1k=1, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When WW is the symmetric group, we obtain the poset of classical kk-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to related background material. Before defining our generalization NC(k)(W)NC^{(k)}(W), we develop from scratch the theory of algebraic noncrossing partitions NC(W)NC(W). This involves studying a finite Coxeter group WW with respect to its generating set TT of {\em all} reflections, instead of the usual Coxeter generating set SS. This is the first time that this material has appeared in one place. Finally, it turns out that our poset NC(k)(W)NC^{(k)}(W) shares many enumerative features in common with the ``generalized nonnesting partitions'' of Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In particular, there is a generalized ``Fuss-Catalan number'', with a nice closed formula in terms of the invariant degrees of WW, that plays an important role in each case. We give a basic introduction to these topics, and we describe several conjectures relating these three families of ``Fuss-Catalan objects''.Comment: Final version -- to appear in Memoirs of the American Mathematical Society. Many small improvements in exposition, especially in Sections 2.2, 4.1 and 5.2.1. Section 5.1.5 deleted. New references to recent wor

    Geometric and Algebraic Combinatorics

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    The 2015 Oberwolfach meeting “Geometric and Algebraic Combinatorics” was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) counterexamples to the topological Tverberg conjecture, and (2) the latest results around the Heron-Rota-Welsh conjecture

    Cluster algebras, quiver representations and triangulated categories

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    This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.Comment: 53 pages, references update

    Cluster ensembles, quantization and the dilogarithm

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    Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its group of symmetries - the cluster modular group, and a relation with the motivic dilogarithm. We define a q-deformation of the X-space. Formulate general duality conjectures regarding canonical bases in the cluster ensemble context. We support them by constructing the canonical pairing in the finite type case. Interesting examples of cluster ensembles are provided the higher Teichmuller theory, that is by the pair of moduli spaces corresponding to a split reductive group G and a surface S defined in math.AG/0311149. We suggest that cluster ensembles provide a natural framework for higher quantum Teichmuller theory.Comment: Version 7: Final version. To appear in Ann. Sci. Ecole Normale. Sup. New material in Section 5. 58 pages, 11 picture

    Aspects of representation theory: τ-exceptional sequences, modular Fuss-Catalan numbers and idempotent completion of extriangulated categories

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    Abstract This thesis is concerned with various aspects of the representation the- ory of finite dimensional algebras, with a focus on combinatorial and homological aspects. We explore the aspects of representation theory relating to tilting modules, cluster algebras, τ-exceptional sequences, and extriangulated categories. The notion of a τ-exceptional sequence was introduced by Buan and Marsh in Buan & Marsh (2021) as a generalisation of an exceptional sequence for finite-dimensional algebras. We calculate the number of complete τ-exceptional sequences of certain classes of Nakayama al- gebras. In some cases, we obtain closed formulas which also count other well-known combinatorial sets and exceptional sequences of path algebras of Dynkin quivers. The modular Catalan numbers C(k,n), introduced in Hein & Huang (2017) count equivalence classes of parenthesizations of x0 ∗ · · · ∗ xn, where ∗ is a binary k-associative operation and k is a positive inte- ger. The classical notion of associativity coincides with 1-associativity, in which case C(1,n) = 1, and the single 1-equivalence class has size given by the Catalan number Cn. We introduce modular Fuss-Catalan numbers Cm which count k-equivalence classes of parenthesizations of k,n x0 ∗ · · · ∗ xn where ∗ is an m-ary k-associative operation for m ≄ 2. Our main results are an explicit formula for Cm , and a characterisation of k-associativity. Extriangulated categories were introduced by Nakaoka and Palu in Nakaoka & Palu (2019a) as a simultaneous generalisation of exact cat- egories and triangulated categories. We show that the idempotent completion of an extriangulated category is also extriangulated. A possible consequence of this is a methodology for constructing Krull- Remak-Schmidt extriangulated categories, since an additive category A has the Krull-Remak-Schmidt property if and only if A is idempotent complete and the endomorphism ring of every object is semi-perfect; see (Krause, 2015, Corollary 4.4)
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