27 research outputs found
The Discrete Fundamental Group of the Associahedron, and the Exchange Module
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 . We also find a combinatorial
description for a basis of this rank. We also introduce the exchange module of
the type 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 .Comment: 16 pages, 4 figure
Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups
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 for each
finite Coxeter group and for each positive integer . When , our
definition coincides with the generalized noncrossing partitions introduced by
Brady-Watt and Bessis. When is the symmetric group, we obtain the poset of
classical -divisible noncrossing partitions, first studied by Edelman.
Along the way, we include a comprehensive introduction to related background
material. Before defining our generalization , we develop from
scratch the theory of algebraic noncrossing partitions . This involves
studying a finite Coxeter group with respect to its generating set of
{\em all} reflections, instead of the usual Coxeter generating set . This is
the first time that this material has appeared in one place.
Finally, it turns out that our poset 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 , 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
The 2015 Oberwolfach meeting âGeometric and Algebraic Combinatoricsâ 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
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
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: Ï-exceptional sequences, modular Fuss-Catalan numbers and idempotent completion of extriangulated categories
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, Ï-exceptional sequences, and extriangulated categories.
The notion of a Ï-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 Ï-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 â · · · â xn,
where â 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 â · · · â xn where â is an m-ary k-associative operation for m â„ 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)