# Tagged Questions

For questions about Coxeter groups, an abstract group that admits a formal description in terms of reflections.

26 views

### I don't know what this symbol in root systems means (of coxeter groups)

I'm reading Humphreys, Reflection groups and Coxeter groups. The section "Construction of root systems" and the books uses the symbol $\mathop {\alpha}\limits^{\sim}$ to denote an special element. ...
16 views

### Existence of Kazhdan Lusztig basis proof due to Soergel

This question is regarding the proof of the existence and uniqueness of Kazhdan Lusztig basis theorem for an arbitrary coxeter group $W$ due to Kazhdan and Lusztig in his paper "Representations of ...
22 views

8 views

### Finding a polytope in the Cartan Subalgebra

The finite Coxeter groups can be realized as symmetry groups of (semi)-regular polytopes. Not all semi-regular polytopes can be realized this way, but all regular polytopes can. Some examples of ...
32 views

### Name of a family of Coxeter groups

From the following image I know that the first of group is the symmetric group of rank $n$ and the second is known as the Hyperoctahedral group. I want to know if someone knows the name of the ...
15 views

### Number of conjugacy classes or bounds for the index of stabilizers on finite Coxeter groups.

For a finite group $G$ denote by $\operatorname{conj}\left(G\right)$ the number of conjugacy classes of $G$ and let $q\left(G\right)$ be the quotient: \begin{align} q\left(G\right)=\frac{\...
20 views

### Why is reflection length equal to codimension of fixed subspace in a real reflection group?

If you have a finite, real reflection group, why can the length function $\ell(-)$ be interpreted as the codimension of the fixed subspace, or alternatively, as the number of eigenvalues different ...
38 views

### Kazhdan-Lusztig polynomials for the longest element in finite Coxeter groups

Suppose $W$ is a finite Coxeter group and that $w_\circ$ is the longest element. I want to prove that $P_{x,w_\circ}=1$ for all $x \in W$. I know from Corollary 7.14 in the book Humphreys p. 167 ...
20 views

### Looking for conditions on generator sets of the form $\{a,b \}$ and $\{a,b,c\}$ on the group $\mathbb{Z}_2^3 \rtimes S_3$

Let $G$ be the group $\mathbb{Z}_2^3 \rtimes S_3$ with the natural action of $S_3$ on the coordinates of $\mathbb{Z}_2^3$. I want to know if there are subsets of $G$ of two elements or also 3 elements ...
14 views

### Connceted componets of a complement of a union of hyperplanes

Let $(W,S)$ be an irreducible coxeter system with graph $\Gamma$ and associated bilinear form $B$. a) $B$ is nondegenerate, but not positive definite. b) For each $s \in S$, the coxeter graph ...
5 views

23 views

### signature of a bilinear form

This question is regarding the proof of a lemma in the book Reflection groups and Coxeter groups by Humphreys section 6.8. Lemma: let $E$ be an n-dimensional real vector space endowed with a ...
32 views

30 views

### If $P_I$ is a parabolic subgroup in $GL_n(k)$, is $P_I$ conjugate to ${P_I}^-$ under some element of $W$?

Suppose $P_I$ is a standard parabolic subgroup of $GL_n(k)$ (we can assume $k$ is finite, but I doubt it matters). Is $P_I$ conjugate to the opposite parabolic ${P_I}^-$ under some element of the Weyl ...
34 views

### Is there a quick way to find the longest element of a subgroup of a Weyl group?

Suppose $W$ is a Weyl group generated by a set of reflections $S$. For $I\subset S$, there are subgroups $W_I$ generated by the reflections $s\in I$. If $w_I$ denotes the longest element of $W_I$, is ...
37 views

85 views

### Centralizers of reflections in parabolic subgroups of Coxeter groups

Let us consider a (not necessarily finite) Coxeter group $W$ generated by a finite set of involutions $S=\{s_1,...,s_n\}$ subject (as usual) to the relations $(s_is_j)^{m_{i,j}}$ with $m_{i,j}=m_{j,i}$...
31 views

### When is a stabilizer group a reflection group?

Let $G$ be a compact, connected Lie group and $K$ a closed, connected subgroup. If $K = T$ is a maximal torus, it is well known that $W := N_G(T)/Z_G(T) = N_G(T)/T$ is a finite reflection group, the ...
78 views

### What decides if a Coxeter Group is “crystalline” or “non-crystalline”?

I am currently writing a research essay concerning Crystallographic theories applied to Virology. Do keep in mind that I am not a mathematician and I know very little about Coxeter groups in general. ...
25 views

### The product of Weyl groups is the Weyl group of the product

Let $G$ and $H$ be compact, connected Lie groups. Write $W(-)$ for the Weyl group. Then it is not hard to see that $$W(G \times H) \cong W(G) \times W(H).$$ Where can I find a published statement of ...
29 views

### An injection of Weyl groups

I've shown, quite accidentally, that Weyl group of $F_4$ injects into the Weyl group of $E_6$ as the subgroup of elements normalizing a maximal torus $T^4$ of $F_4$. One might a priori expect other ...
32 views

### Affine Hecke algebras and Lusztig relations

I study the book "Affine Hecke algebras and orthogonal polynomials" by I.G. Macdonald. He propose a formula in section $4.2$, especially formula $(4.2.9)$. This formula is the following: T(w)f(Y)=\...
123 views

### Weyl groups: correspondence of reflections and roots?

If W is the Weyl group of some ADE-type Lie algebra, and w is an element corresponding to a reflection (not just an involution), does it necessarily correspond to a root?
82 views

### Isomorphic Coxeter Groups

Is it true that two Coxeter groups having the same Coxeter matrix (equivalently, the same Coxeter graph) isomorphic? Because otherwise, the definition of a Coxeter group from its Coxeter matrix does ...
43 views

### Use GAP3 program for computing the h-polynomials of Lusztig

I am using GAP to compute things in Kazhdan-Lusztig theory, especially using the package "Chevie". According to the GAP3 Manual, the base change between the "C"-basis and "T"-basis can be computed, ...
51 views

### Texts on Coxeter groups

I'm looking for an introductory text on Coxeter groups. It can assume undegraduate knowledge of Algebra (Groups up to and including the Sylow theorems in Fraleigh, elementary knowledge of rings, ...
26 views

### Lusztig's $h$-function of a dihedral group

Following the notations in Hecke algebras with unequal parameters, let $(W,S,L)$ be a weighted Coxeter system, and $H$ be the corresponding Hecke algebra with $\{c_w |w \in W\}$ the Kazhdan-Lusztig ...
43 views

### Create Platonic solids from the coxeter group (vertexes & edges & faces)

How can one define vertexes, edges and faces from the Coxeter group? For example, for all platonic solids? I would like to create a general function that takes the Coxeter diagram as input, and gives ...
39 views

### Simple systems are the “smallest”

Let $\Phi$ be a root system of a finite reflection group $W$. Let $\Delta$ be a simple system in $\Phi$. I want to prove that $\Delta$ is "the smallest" set which generates $W$, more precisely: There ...
297 views

### Is every finite group of isometries a subgroup of a finite reflection group?

Is every finite group of isometries in $d$-dimensional Euclidean space a subgroup of a finite group generated by reflections? By "reflection" I mean reflection in a hyperplane: the isometry fixing a ...
95 views

### Is every finitely generated Kleinian group commensurable to a Coxeter group?

Or, to a finitely generated reflection group? Here, I do not insist that the Coxeter group is represented as a hyperbolic reflection group. If not, what is an counterexample? And what is a ...