A central limit theorem on two-sided descents of Mallows distributed elements of finite Coxeter groups

A note on involution prefixes in Coxeter groups

Abstract The Kac-Moody affine Hecke algebra $$\mathcal {H}$$ H was first constructed as the Iwahori-Hecke algebra of a p -adic Kac-Moody group by work of Braverman, Kazhdan, and Patnaik, and by work of Bardy-Panse, Gaussent, and Rousseau. Since $$\mathcal {H}$$ H has a Bernstein presentation, for affine types it is a positive-level variation of Cherednik’s double affine Hecke algebra. Moreover, as $$\mathcal {H}$$ H is realized as a convolution algebra, it has an additional “ T -basis” corresponding to indicator functions of double cosets. For classical affine Hecke algebras, this T -basis reflects the Coxeter group structure of the affine Weyl group. In the Kac-Moody affine context, the indexing set $$W_{\mathcal {T}}$$ W T for the T -basis is no longer a Coxeter group. Nonetheless, $$W_{\mathcal {T}}$$ W T carries some Coxeter-like structures: a Bruhat order, a length function, and a notion of inversion sets. This paper contains the first steps toward a Coxeter theory for Kac-Moody affine Hecke algebras. We prove three results. The first is a construction of the length function via a representation of $$\mathcal {H}$$ H . The second concerns the support of products in classical affine Hecke algebras. The third is a characterization of length deficits in the Kac-Moody affine setting via inversion sets. Using this characterization, we phrase our support theorem as a precise conjecture for Kac-Moody affine Hecke algebras. Lastly, we give a conjectural definition of a Kac-Moody affine Demazure product via the $$q=0$$ q = 0 specialization of $$\mathcal {H}$$ H .

Let n ≥ 3 , and let Out ( W n ) be the outer automorphism group of a free Coxeter group W n of rank n . We study the growth of the dimension of the homology groups (with coefficients in any field 𝕂 ) along Farber sequences of finite-index subgroups of Out ( W n ) . We show that, in all degrees up to ⌊ n 2 ⌋ - 1 , these Betti numbers grow sublinearly in the index of the subgroup. When 𝕂 = ℚ , through Lück’s approximation theorem, this implies that all ℓ 2 -Betti numbers of Out ( W n ) vanish up to degree ⌊ n 2 ⌋ - 1 . In contrast, in top dimension equal to n - 2 , an argument of Gaboriau and Noûs implies that the ℓ 2 -Betti number does not vanish. We also prove that the torsion growth of the integral homology is sublinear. Our proof of these results relies on a recent method introduced by Abért, Bergeron, Frączyk and Gaboriau. A key ingredient is to show that a version of the complex of partial bases of W n has the homotopy type of a bouquet of spheres of dimension ⌊ n 2 ⌋ - 1 .

Abstract In a short paper that appeared in the Journal of the London Mathematical Society in 1934, H. S. M. Coxeter completed the classification of finite Coxeter groups. In this survey, we describe what Coxeter did in this paper and examine an assortment of topics that illustrate the broad and enduring influence of Coxeter's paper on developments in algebra, group theory, and geometry.

Abstract The existence of acyclic complete matchings on the face poset of a regular CW complex implies that the underlying topological space of the CW complex is contractible by discrete Morse theory. In this paper, we construct explicitly acyclic complete matchings on any nontrivial Bruhat interval based on any reflection order on the Coxeter group . We then apply this combinatorial result to regular CW complexes arising from the theory of total positivity. As an application, we show that the totally nonnegative Springer fibers are contractible. This verifies a conjecture of Lusztig. As another application, we show that the totally nonnegative fibers of the natural projection from full flag varieties to partial flag varieties are contractible. This leads to a much simplified proof of the regularity property on totally nonnegative partial flag varieties compared to the proofs by Galashin et al. and Bao and He.

We discuss examples of linear representations of finite groups as subgroups of the Riordan group. In particular, we show that the symmetric group of degree three has no faithful representation as a subgroup of the Riordan group over the complex numbers, but can be embedded as a subgroup of the Riordan group over a field of characteristic three.

A group has Property (NL) if it does not admit a loxodromic element in any hyperbolic action. In other words, a group with this property is inaccessible for study from the perspective of hyperbolic actions. This property was introduced by Balasubramanya, Fournier-Facio and Genevois, who initiated the study of this property. We expand on this research by studying Property (NL) in Coxeter groups, a class of groups that are defined by an underlying graph. One of our main results show that a right-angled Coxeter group (RACG) has Property (NL) if and only if its defining graph is complete. We then move beyond the right-angled case to show that if a defining graph is disconnected, its corresponding Coxeter group does not have Property (NL). Lastly, we classify which triangle groups (Coxeter groups with three generators) have Property (NL).

A threshold for relative hyperbolicity in random right-angled Coxeter groups

ABSTRACT A finite group is mixable if a product of random elements, each chosen independently from two options, can distribute uniformly on . We present conditions and obstructions to mixability. We show that 2‐groups, the symmetric groups, the simple alternating groups, several matrix and sporadic simple groups, and most finite Coxeter groups, are mixable. We also provide bounds on the mixing length of such groups.

In this paper, we study properties of random walks on finite groups and later use them to obtain the limiting braid length expectation and component number of braid closure in a model of random braids, which is constructed by lifting elements of random walk on a Coxeter group to a braid group.

Here, for $W$ the Coxeter group $\mathrm{D}_n$ where $n > 4$, it is proved that the maximal rank of an abstract regular polytope for $W$ is $n - 1$ if $n$ is even and $n$ if $n$ is odd. Further it is shown that $W$ has abstract regular polytopes of rank $r$ for all $r$ such that $3 \leq r \leq n - 1$, if $n$ is even, and $3 \leq r \leq n$, if $n$ is odd. The possible ranks of abstract regular polytopes for the exceptional finite irreducible Coxeter groups are also determined.

We classify two-dimensional right-angled Coxeter groups that are quasiisometric to a right-angled Artin group defined by a tree, and show that when this is true the right-angled Coxeter group actually contains a visible finite index right-angled Artin subgroup.

The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection arrangements and their deformations. Inspired by the recent work of Bernardi, we show that the notion of moves and sketches can be used to provide a uniform and explicit bijection between regions of (the Catalan deformation of) a reflection arrangement and certain non-nesting partitions. We then use the exponential formula to describe a statistic on these partitions such that distribution is given by the coefficients of the characteristic polynomial. Finally, we consider a sub-arrangement of type C arrangement called the threshold arrangement and its Catalan and Shi deformations.

High dimensional hyperbolic Coxeter groups that virtually fiber

We describe the geometry of conjugation within any split subgroup H of the full isometry group G of n -dimensional Euclidean space. We prove that, for any h \in H , the conjugacy class [h]_{H} of h is described geometrically by the move-set of its linearization, while the set of elements conjugating h to a given h'\in [h]_{H} is described by the fix-set of the linearization of h' . Examples include all affine Coxeter groups, certain crystallographic groups, and the group G itself.

Coxeter groups are biautomatic

Abstract Brick polytopes constitute a remarkable family of polytopes associated to the spherical subword complexes of Knutson and Miller. They were introduced for finite Coxeter groups by Pilaud and Stump, who used them to produce geometric realizations of generalized associahedra arising from the theory of cluster algebras of finite types. In this paper, we present an application of the vast generalization of brick polyhedra for general subword complexes (not necessarily spherical) recently introduced by Jahn and Stump. More precisely, we show that the $$\nu $$ ν -associahedron, a polytopal complex whose edge graph is the Hasse diagram of the $$\nu $$ ν -Tamari lattice introduced by Préville-Ratelle and Viennot, can be geometrically realized as the complex of bounded faces of the brick polyhedron of a well chosen subword complex. We also present a suitable projection to the appropriate dimension, which leads to an elegant vertex-coordinate description.

The deletion order and Coxeter groups

Abstract For a Coxeter group 𝑊 with length function ℓ, the excess zero graph E 0 ⁢ ( W ) \mathcal{E}_{0}(W) has vertex set the non-identity involutions of 𝑊, with two involutions 𝑥 and 𝑦 adjacent whenever ℓ ⁢ ( x ⁢ y ) = ℓ ⁢ ( x ) + ℓ ⁢ ( y ) \ell(xy)=\ell(x)+\ell(y) . Properties of this graph such as connectivity, diameter and valencies of certain vertices of E 0 ⁢ ( W ) \mathcal{E}_{0}(W) are explored.