Counting nearest faraway flats for Coxeter chambers

We point out that the moduli spaces of all known 3d mathcal{N} = 8 and mathcal{N} = 6 SCFTs, after suitable gaugings of finite symmetry groups, have the form ℂ4r/Γ where Γ is a real or complex reflection group depending on whether the theory is mathcal{N} = 8 or mathcal{N} = 6, respectively. Real reflection groups are either dihedral groups, Weyl groups, or two sporadic cases H3,4 Since the BLG theories and the maximally supersymmetric Yang-Mills theories correspond to dihedral and Weyl groups, it is strongly suggested that there are two yet-to­be-discovered 3d mathcal{N} = 8 theories for H3,4. We also show that all known mathcal{N} = 6 theories correspond to complex reflection groups collectively known as G(k, x, N). Along the way, we demonstrate that two ABJM theories (SU(N)k x SU(N)-k)/ℤN and (U(N)k x U(N)-k) /ℤk are actually equivalent.

Perfect models for finite Coxeter groups

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ . We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ .

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

We classify surjective lattice homomorphisms $W\to W'$ between the weak orders on finite Coxeter groups. Equivalently, we classify lattice congruences $\Theta$ on $W$ such that the quotient $W/\Theta$ is isomorphic to $W'$. Surprisingly, surjective homomorphisms exist quite generally: They exist if and only if the diagram of $W'$ is obtained from the diagram of $W$ by deleting vertices, deleting edges, and/or decreasing edge labels. A surjective homomorphism $W\to W'$ is determined by its restrictions to rank-two standard parabolic subgroups of $W$. Despite seeming natural in the setting of Coxeter groups, this determination in rank two is nontrivial. Indeed, from the combinatorial lattice theory point of view, all of these classification results should appear unlikely a priori. As an application of the classification of surjective homomorphisms between weak orders, we also obtain a classification of surjective homomorphisms between Cambrian lattices and a general construction of refinement relations between Cambrian fans.

This thesis presents recent developments concerning two open problems related to algebra and discrete geometry. In 1934, Harold S. M. Coxeter introduced Coxeter groups as abstractions of groups generated by reflections in a vector space [Cox34]. The present work lays out how a study of geometric and combinatorial properties of Coxeter groups contributed to the comprehension of the two open problems. Furthermore, this thesis presents some progress in proving the first problem, and an application of the approach introduced in this thesis for the second problem. Open Problem (Lattice for infinite Coxeter groups, Dyer [Dye11]). Is there, for each infinite Coxeter group, a complete ortholattice that contains the weak order? In Chapter 2, we study the asymptotical behaviour of roots of infinite Coxeter groups, which is part of a joint work with Christophe Hohlweg and Vivien Ripoll [HLR13]. In particular, we show that the directions of roots tend to the isotropic cone of the geometric representation of the root system. Moreover, using this framework, this thesis presents a proof that there is a complete ortholattice structure enclosing the weak order of infinite Coxeter groups of rank at most 3. Open Problem (Existence of multiassociahedra, Jonsson [Jon05]). Let k be an integer such that k >= 1. Is there a polytope whose boundary complex corresponds to the simplicial complex of sets of diagonals of a convex polygon not containing k + 1 mutually crossing diagonals? In Chapter 3 we introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. This family generalizes the concept of multitriangulations of type A and B to arbitrary finite Coxeter groups. For k = 1, this simplicial complex coincides with the finite cluster complex of the given type. We study combinatorial and geometric properties of multi-cluster complexes. In particular, we show that every spherical subword complex is the link of a face of a multi-cluster complex. This work was realized jointly with Cesar Ceballos and Christian Stump [CLS13]. Finally, this approach allows us to exhibit formulas counting the number of common vertices of permutahedra and generalized associahedra for arbitrary finite Coxeter groups and Coxeter elements.

The complement of the hyperplane arrangement associated to the (complexified) action of a finite, real reflection group on \mathbb C^n is known to be a {\mathcal K}(\pi ,1) space for the corresponding Artin group \mathcal A . A long-standing conjecture states that an analogous statement should hold for infinite reflection groups. In this paper we consider the case of a Euclidean reflection group of type \tilde{\mathcal A}_n and its associated Artin group, the affine braid group \tilde{\mathcal A} . Using the fact that \tilde{\mathcal A} can be embedded as a subgroup of a finite type Artin group, we prove a number of conjectures about this group. In particular, we construct a finite, n -dimensional {\mathcal K}(\pi ,1) -space for \tilde{\mathcal A} , and use it to prove the {\mathcal K}(\pi ,1) -conjecture for the associated hyperlane complement. In addition, we show that the affine braid groups are biautomatic and give an explicit biautomatic structure.

Vanishing ranges for the mod p cohomology of alternating subgroups of Coxeter groups

Reflection groups of geodesic spaces and Coxeter groups

On excess in finite Coxeter groups

Modular elements in the lattice L(A) when A is a real reflection arrangement

Cambrian lattices

A lattice isomorphism theorem for cluster groups of mutation-Dynkin type An

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript down-arrow Superscript c"> <mml:semantics> <mml:msubsup> <mml:mi> π </mml:mi> <mml:mo stretchy="false"> ↓ </mml:mo> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\pi _\downarrow ^c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mapping each element of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W"> <mml:semantics> <mml:mi>W</mml:mi> <mml:annotation encoding="application/x-tex">W</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to the unique maximal <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sortable element below it in the weak order. The fibers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript down-arrow Superscript c"> <mml:semantics> <mml:msubsup> <mml:mi> π </mml:mi> <mml:mo stretchy="false"> ↓ </mml:mo> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\pi _\downarrow ^c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> essentially define the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c"> <mml:semantics> <mml:mi>c</mml:mi> <mml:annotation encoding="application/x-tex">c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi Subscript down-arrow Superscript c"> <mml:semantics> <mml:msubsup> <mml:mi> π </mml:mi> <mml:mo stretchy="false"> ↓ </mml:mo> <mml:mi>c</mml:mi> </mml:msubsup> <mml:annotation encoding="application/x-tex">\pi _\downarrow ^c</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.

Let W W be an infinite irreducible Coxeter group with ( s 1 , … , s n ) (s_1, \ldots , s_n) the simple generators. We give a short proof that the word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s n ⋯ s 1 s 2 ⋯ s n s_n \cdots s_1 s_2 \cdots s_n is reduced for any number of repetitions of s 1 s 2 ⋯ s n s_1 s_2 \cdots s_n . This result was proved for simply laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof uses only basic facts about Coxeter groups and the geometry of root systems. We also prove that, in finite Coxeter groups, there is a reduced word for w 0 w_0 which is obtained from the semi-infinite word s 1 s 2 ⋯ s n s 1 s 2 ⋯ s n ⋯ s_1 s_2 \cdots s_n s_1 s_2 \cdots s_n \cdots by interchanging commuting elements and taking a prefix.