8
votes
0answers
191 views

An angle-doubling trick of Kirillov and Berenstein [on hold]

Kirillov and Berenstein, in their article "Groups generated by involutions, Gelfand-Tsetlin patterns and combinatorics of Young tableaux" (available at math.uoregon.edu/~arkadiy/bk1.pdf), present a ...
2
votes
1answer
76 views

Double quotients of Coxeter groups have the chain property?

Let $(W,S)$ be a Coxeter group with Bruhat order $\leq$ and length function $\ell(w)$. Definition: a subset $X \subset W$ has the chain property if whenever $x,y \in X$ with $x < y$, there ...
12
votes
3answers
498 views

Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram ...
15
votes
2answers
403 views

Is there a natural notion of completion of a Coxeter system?

Let $(W,S)$ be a Coxeter system. Then any element of $W$ can be written as a finite products of elements of $S$. I want some notion of a "completion" of $W$, call it $\hat{W}$, whose elements are ...
2
votes
1answer
158 views

Which subgroups of a finite reflection group have distingushed coset representatives?

Let $W$ be a finite reflection group with length function $l$ and let $I$ be a set of simple reflections that generate $W$. Let $\phi$ be an automorphism of $W$ permuting $I$. Consider the orbits of ...
19
votes
0answers
246 views

Uniform proof that a finite reflection group is determined by its degrees?

Given a finite (real) group $G$ generated by reflections acting on euclidean $n$-space, it was shown by Chevalley in the 1950s that the algebra of invariants of $G$ in the associated polynomial ...
6
votes
1answer
353 views

Quotient of Coxeter group

Since the group $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{10} \rangle$ seems to have resisted attacks of some powerful programs, I will turn to a group that seems to be a bit easier to ...
6
votes
1answer
240 views

Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs?

There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl ...
25
votes
3answers
755 views

Is there any need to study Coxeter systems (W,S) with S infinite?

In their treatise Groupes et algebres de Lie, Bourbaki (no doubt heavily influenced by Tits) devoted Chapter IV (1968) to the general theory of what they dubbed "Coxeter systems" $(W,S)$ along with ...
6
votes
2answers
136 views

Irreducibility of Coxeter Graphs as a Function of Generating Sets

Given a Coxeter system $(W, S)$, we can form its Coxeter graph, and say that the system is irreducible if the graph is connected. Now, irreducibility is not solely a function of $W$; it depends also ...
11
votes
2answers
757 views

Random walks on Coxeter groups

Let $G_N$ be the group generated by elements $a_1,\ldots,a_N$ subject to the relations $a_i^2=1$ and $(a_ia_j)^3=1$. The growth function of $G_N$ is then ...
2
votes
2answers
421 views

elements in the weyl group

Let W be the Weyl a group of a semisimple simply connected group over C. Let I={1,...,r} the set of simple roots. For $w\in W$, I denote by supp(w) the subset of I corresponding to the simple ...
6
votes
2answers
435 views

Algorithm for reducing words in a Coxeter group

Let $W$ be a Coxeter group with set of simple reflections $S$. Suppose that I have chosen a preferred reduced decomposition for every element of $W$. Given an arbitrary word in the alphabet $S$, is ...
9
votes
3answers
390 views

Are there Hamilton paths in Cayley graphs of Coxeter groups?

Hi everyone. I want to optimize certain computation on finite Coxeter groups $(W,S)$. Basically I compute the matrices $\rho(T_w)$ for all $w\in W$ of a matrix representation $H\to K^{d\times d}$ of ...
5
votes
4answers
562 views

Infinite Coxeter groups with a non-trivial finite conjugacy class?

Let $(W,S)$ be a Coxeter system, where $S$ is finite. Assume that $W$ has an infinite number of elements. Is it true that conjugacy classes of elements of non-central elements of $S$ have always an ...
4
votes
1answer
162 views

When do reflection groups act discretely on quadrics in indefinite/semi-Riemannian situation?

The hyperbolic case seems to be well understood after work of Vinberg. Given a lattice $L$ with quadratic form $Q$ of signature $(1,n)$, the orthogonal group of $L$ acts discretely on the affine ...
9
votes
0answers
333 views

Reflection groups in O(n+1,n) arising `in nature'?

For a while a friend and I have been thinking about a family of integral symmetric bilinear forms of signature $(n+1,n)$. Such lattices in our case arise 'in nature' (in a certain problem about vector ...
12
votes
2answers
615 views

Does the Poincare series of a Coxeter group always describe a “flag variety”?

Let $W$ be a Coxeter group and let $P_W(q) = \sum_{w \in W} q^{\ell(w)}$ be its Poincare series. When $W$ is the Weyl group of a simple algebraic group $G$ (hence $W$ is finite), $P_W(q)$ is the ...