Coxeter element
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K(π, 1) for Artin Groups of Finite Type Open
This paper is a continuation of a programme to construct new K(π, 1)’s for Artin groups of finite type which began in [4] with Artin groups on 2 and 3 generators and was extended to braid groups in [3]. These K(π, 1)’s differ from those in…
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Resurgence and dynamics of O(N) and Grassmannian sigma models Open
We study the non-perturbative dynamics of the two dimensional O(N ) and Grassmannian sigma models by using compactification with twisted boundary conditions on $$ \mathbb{R}\times {S}^1 $$ , semi-classical techniques and resurgence. While …
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Birational automorphism groups and the movable cone theorem for Calabi-Yau manifolds of Wehler type via universal Coxeter groups Open
Thanks to the theory of Coxeter groups, we produce the first family of Calabi-Yau manifolds $X$ of arbitrary dimension $n$, for which ${\rm Bir}(X)$ is infinite and the Kawamata-Morrison movable cone conjecture is satisfied. For this famil…
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Homological stability for families of Coxeter groups Open
We prove that certain families of Coxeter groups and inclusions\n$W_1\\hookrightarrow W_2\\hookrightarrow...$ satisfy homological stability,\nmeaning that in each degree the homology $H_\\ast(BW_n)$ is eventually\nindependent of $n$. This …
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Deligne-Lusztig theoretic derivation for Weyl groups of the number of reflection factorizations of a Coxeter element Open
Chapuy and Stump have given a nice generating series for the number of factorizations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to …
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Toggling Independent Sets of a Path Graph Open
This paper explores the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove a generalization of a homomes…
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Simple transitive 2‐representations of Soergel bimodules for finite Coxeter types Open
In this paper, we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive 2‐representations and we complete their classification in all types but and .
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Pop-stack-sorting for Coxeter groups Open
Let \(W\) be an irreducible Coxeter group. We define the Coxeter pop-stack-sorting operator \(\mathsf{Pop}:W\to W\) to be the map that fixes the identity element and sends each nonidentity element \(w\) to the meet of the elements covered …
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A categorification of non-crossing partitions Open
We present a categorification of the non-crossing partitions given by crystallographic Coxeter groups. This involves a category of certain bilinear lattices, which are essentially determined by a symmetrisable generalised Cartan matrix tog…
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Quasi-Coxeter quasitriangular quasibialgebras and the Casimir connection Open
Let g be a complex, semisimple Lie algebra. We prove the existence of a quasi-Coxeter, quasitriangular quasibialgebra structure on the enveloping algebra of g, which binds the quasi-Coxeter structure underlying the Casimir connection of g …
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On enumerating factorizations in reflection groups Open
We describe an approach, via Malle’s permutation Ψ on the set of irreducible characters Irr(W) of a reflection group W, that gives a uniform derivation of the Chapuy–Stump formula for the enumeration of reflection factorizations of a Coxet…
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Coxeter groups, quiver mutations and geometric manifolds Open
We construct finite volume hyperbolic manifolds with large symmetry groups. The construction makes use of the presentations of finite Coxeter groups provided by Barot and Marsh, and involves mutations of quivers and diagrams defined in the…
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Poset edge densities, nearly reduced words, and barely set-valued tableaux Open
In certain finite posets, the expected down-degree of their elements is the same whether computed with respect to either the uniform distribution or the distribution weighting an element by the number of maximal chains passing through it. …
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Weak generalized lifting property, Bruhat intervals, and Coxeter matroids Open
We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also de…
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On cycle decompositions in Coxeter groups Open
The aim of this note is to show that the cycle decomposition of elements of the symmetric group admits a quite natural formulation in the framework of dual Coxeter theory, yielding a generalization of it to the family of so-called paraboli…
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Hecke action on the principal block Open
In this paper we construct an action of the affine Hecke category (in its ‘Soergel bimodules’ incarnation) on the principal block of representations of a simply connected semisimple algebraic group over an algebraically closed field of cha…
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Coxeter combinatorics and spherical Schubert geometry Open
For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are…
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Noncrossing partitions, Bruhat order and the cluster complex Open
We introduce two order relations on finite Coxeter groups which refine the absolute and the Bruhat order, and establish some of their main properties. In particular, we study the restriction of these orders to noncrossing partitions and sh…
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Applications of geometric techniques in Coxeter-Catalan combinatorics Open
University of Minnesota Ph.D. dissertation. September 2017. Major: Mathematics. Advisor: Victor Reiner. 1 computer file (PDF); ix, 95 pages.
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A simple character formula Open
In this paper we prove a character formula expressing the classes of simple representations in the principal block of a simply-connected semisimple algebraic group in terms of baby Verma modules, under the assumption that the characterist…
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Noncrossing partitions of an annulus Open
The noncrossing partition poset associated to a Coxeter group $W$ and Coxeter element $c$ is the interval $[1,c]_T$ in the absolute order on $W$. We construct a new model of noncrossing partititions for $W$ of classical affine type, using …
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A Refined Count of Coxeter Element Reflection Factorizations Open
For well-generated complex reflection groups, Chapuy and Stump gave a simple product for a generating function counting reflection factorizations of a Coxeter element by their length. This is refined here to record the numberof reflections…
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A two-sided analogue of the Coxeter complex Open
For any Coxeter system (W, S) of rank n, we introduce an abstract boolean complex (simplicial poset) of dimension 2n − 1 which contains the Coxeter complex as a relative subcomplex. Faces are indexed by triples (J,w,K), where J and K are s…
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Cluster realizations of Weyl groups and higher Teichmüller theory Open
For a symmetrizable Kac-Moody Lie algebra $\mathfrak{g}$, we construct a family of weighted quivers $Q_m(\mathfrak{g})$ ($m \geq 2$) whose cluster modular group $Γ_{Q_m(\mathfrak{g})}$ contains the Weyl group $W(\mathfrak{g})$ as a subgrou…
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Twisted Coxeter elements and folded AR-quivers via Dynkin diagram automorphisms: I Open
We introduce and study the twisted adapted $r$-cluster point and its combinatorial Auslander-Reiten quivers, called twisted AR-quivers and folded AR-quivers, of type $A_{2n+1}$ which are closely related to twisted Coxeter elements and the …
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On loop Deligne--Lusztig varieties of Coxeter-type for inner forms of ${\rm GL}_n$ Open
For a reductive group $G$ over a local non-archimedean field $K$ one can mimic the construction from the classical Deligne--Lusztig theory by using the loop space functor. We study this construction in special the case that $G$ is an inner…
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Circuits and Hurwitz action in finite root systems Open
In a finite real reflection group, two factorizations of a Coxeter element into an arbitrary number of reflections are shown to lie in the same orbit under the Hurwitz action if and only if they use the same multiset of conjugacy classes. …
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Grothendieck's Dessins d'Enfants in a Web of Dualities. II Open
We show that the spectral curve for Eynard-Orantin topological recursions satisfied by counting Grothendieck's dessins d'enfants are related to Narayana numbers. This suggests a connection of dessins to combinatorics of Coxeter groups, non…
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Lattice bijections for string modules, snake graphs and the weak Bruhat order Open
In this paper we introduce abstract string modules and give an explicit bijection between the submodule lattice of an abstract string module and the perfect matching lattice of the corresponding abstract snake graph. In particular, we make…
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On conjugacy of abstract root bases of root systems of Coxeter groups Open
We introduce and study a combinatorially defined notion of the root basis of a (real) root system of a possibly infinite Coxeter group. Known results on conjugacy up to sign of root bases of certain irreducible finite rank real root system…