Related papers: The boundness of weighted Coxeter groups of rank 3

We prove that there exists a bound $N'_L(W)$ for a positively weighted Coxeter group $(W, S, L)$ of finite rank. In particular, Lusztig's $\boldsymbol{a}$-function of $(W, S, L)$ is bounded.

We show that Lusztig's $a$-function of a Coxeter group is bounded if the rank of the Coxeter group is 3.

Let W be a Coxeter group with Coxeter generators S. The rank of the Coxeter system (W,S) is the cardinality |S| of S. The Coxeter system (W,S) has finite rank if and only if W is finitely generated. If (W,S) has infinite rank, then |S| =…

We show that Lusztig's $a$-function of a Coxeter group is bounded if the Coxeter group has a complete graph (i.e. any two vertices are joined) and the cardinalities of finite parabolic subgroups of the Coxeter group have a common upper…

Given a Coxeter system $(W,S)$ and a multiparameter $\mathbf{q}$ of real numbers indexed by $S$, one can define the weighted $L^2$-cohomology groups and associate to them a nonnegative real number called the weighted $L^2$-Betti number. We…

We prove Lusztig's conjectures P1-P15 for hyperbolic Coxeter groups of rank 3. Our proof enables us to give a description of the a-function and Kazhdan-Lusztig cells for these Coxeter groups.

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…

In this paper, we show that the boundary $\partial\Sigma(W,S)$ of a right-angled Coxeter system $(W,S)$ is minimal if and only if $W_{\tilde{S}}$ is irreducible, where $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in…

In this paper, we investigate boundaries of parabolic subgroups of Coxeter groups. Let $(W,S)$ be a Coxeter system and let $T$ be a subset of $S$ such that the parabolic subgroup $W_T$ is infinite. Then we show that if a certain set is…

If S and S' are two finite sets of Coxeter generators for a right-angled Coxeter group W, then the Coxeter systems (W,S) and (W,S') are equivalent.

In this paper, we give a new class of rigid Coxeter groups. Let $(W,S)$ be a Coxeter system. Suppose that (0) for each $s,t\in S$ such that $m(s,t)$ is even, $m(s,t)=2$, (1) for each $s\neq t\in S$ such that $m(s,t)$ is odd, $\{s,t\}$ is a…

Let W be a Coxeter group and L be a weight function on W. Following Lusztig, we have a corresponding decomposition of W into left cells, which have important applications in representation theory. We study the case where $W$ is an affine…

If $(W,S)$ is a right-angled Coxeter system and $W$ has no $\mathbb Z^3$ subgroups, then it is shown that the absence of an elementary separation property in the presentation diagram for $(W,S)$ implies all CAT(0) spaces acted on…

We study the boundedness of a mutation class for quivers with real weights. The main result is a characterization of bounded mutation classes for real quivers of rank 3.

In this paper, we give a new class of rigid Coxeter groups. Let $(W,S)$ be a Coxeter system. Suppose that (0) for each $s,t\in S$ such that $m(s,t)$ is even, $m(s,t)\in\{2\}\cup 4\N$, (1) for each $s\neq t\in S$ such that $m(s,t)$ is odd,…

Let $(W,S)$ be a Coxeter system and $\Gamma$ be a group of automorphisms of $W$ such that $\gamma(S)=S$ for all $\gamma \in \Gamma$. Then it is known that the group of fixed points $W^\Gamma$ is again a Coxeter group with a canonically…

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special…

The notion of limit roots of a Coxeter group W was recently introduced (see arXiv:1112.5415 and arXiv:1303.6710): they are the accumulation points of directions of roots of a root system for W. In the case where the root system lives in a…

In this paper, we give precise description for the lowest lowest two-sided cell c_{0} and the left cells in it for a weighted Coxeter group of rank 3. Then we show conjectures P1-P15 and \widetilde{P} hold for c_{0} and do some calculation…

Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits…