# Questions tagged [reflection-groups]

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• 1,671
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### Have the affine simplicial line arrangments been enumerated?

I am looking for a classification (or attempt at enumeration) of affine simplicial line arrangements. A line arrangment is a family of straight lines in $\Bbb R^2$. It is simplicial if all regions are ...
• 12.6k
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### Real-isability of a (relatively small) subconfiguration of the Klein configuration

The Klein configuration consists of $60$ points and $60$ planes in $\mathbb C\mathbf P^3$, each point lying on $15$ of the planes and each plane containing $15$ of the points. It appears, among many ...
• 17.6k
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### Invariants for the Weyl group of $\mathrm{SO}_{2n}$ acting on a certain group scheme

Let $W$ denote the Weyl group of $\mathrm{SO}_{2n}$, so $W = (\mathbf{Z}/2)^{n-1} \rtimes \Sigma_n$. There is a natural action of $W$ on $\mathbf{Z}^n$ given by permutations and even numbers of sign ...
• 5,560
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### When are groups generated by reflections in a triangle discrete?

Take a triangle in the (Euclidean or hyperbolic) plane, and consider the group of isometries generated by the reflections in the three sides of the triangle. If the angles between adjacent sides are ...
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• 11
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### Are there infinite abelian real reflection groups?

Lately I have been studying reflection groups, and there is a particular example of a complex reflection group that has been very good for guiding my intuition. I would like to know if there is an ...
215 views

### Classification of octonionic reflection groups

I know that there exist classification theorems for real, complex, and quaternionic, reflection groups. There are presentations for the real reflection groups, as well as further presentations for the ...
• 113
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### Algorithm to determine if a vector in the geometric representation of a Coxeter group is proportional to a root

Let $W$ be a Coxeter group, and let $V$ be its geometric representation (as defined for instance in Section 5.3 of Humphreys' book Reflection groups and Coxeter groups). Let $v\in V\backslash\{0\}$ (...
• 484
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• 2,693
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### Is there a classification of reflection groups over division rings?

I asked a version of this question in Math StackExchange about a week ago but I've received no feedback so far, so following the advice I received on meta I decided to post it here. Details The ...
• 976
311 views

### Generalized root systems and reflection groups

Consider the following alternative definition of finite reflection group: Definition: A finite reflection group $\Gamma\subset\mathrm O(\Bbb R^d)$ is a finite group generated by orthogonal ...
• 12.6k
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### On Shephard-Todd theorem

There is an excellent Torsten Ekedahl's answer to Roman Fedorov's question here: Chevalley–Shephard–Todd theorem. Does anyone know any articles or books where this approach is outlined? I didn't ...
406 views

### How are reflection groups related to general point groups?

I always tried to understand how the finite reflection groups of $\Bbb R^d$ (of some fixed dimension $d$) relate to the point groups of the same space $\smash{\Bbb R^d}$ (finite subgroup of the ...
• 12.6k
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### Reflection reverses a root string

I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root string is at most 4: Theorem If $\alpha,\beta$ are roots ...
• 225
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### Fake degrees: why coinvariant algebra and classical groups over finite fields?

Apologies if this is not research level math (in that it concerns well-known stuff), but I am having trouble tracking down sources that explain the following. References would be very appreciated. ...
• 23k
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### The growth of maximum elements for the reflection group $D_n$

Let's consider maximum coefficients in Poincaré polynomial (or growth series) for the reflection group (or Weyl group, or finite Coxeter group) $D_n$ as mentioned in A162206. The maximal numbers $M(n)$...
301 views

### Coxeter groups generated by one finite conjugacy class

Let $(W,S)$ be an arbitrary Coxeter system. We consider the following scenario: Let $\mathcal{O}$ be a conjugacy class of an element $w$ in $W$ which is finite and which generates the whole group $W$....
• 1,064
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### Basis of coinvariant algebra on which reflection group acts as regular representation

This question is almost a duplicate of a question of Christian Stump, except that Christian seems to ask about an isomorphism to irreducible representations rather than the regular representation: ...
• 23k
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### What is the relation between Coxeter transformations of Coxeter systems and Coxeter transformations of generalized Cartan matrices?

This question is related to the following question about Coxeter transformations that I asked and recently answered myself. For completeness I also write full definitions in the new question. The ...
• 253
464 views

### About reflections of reflection groups

For any finite crystallographic reflection group $W = \langle s_1, \ldots , s_n\rangle$, every hyperplane reflection is of the form $ws_iw^{-1}$ for some $i$ and some $w \in W$. A finite ...
• 331
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### Generalization of cycle decomposition to Coxeter groups

I'm looking for a generalization of cycle decompositions for permutations to elements of Coxeter groups. (For the purposes of this question, any conjugate of a parabolic subgroup is also a parabolic ...
• 2,868
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### Roots of exceptional complex reflection groups

I am looking to do a case-by-case check of a conjecture I have about Shephard groups, which are a subclass of complex reflection groups. These were classified by Shephard and Todd and there is one ...
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• 351
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### Generate harmonic polynomials for a finite group

Let $G$ be a finite group acting on a complex vector space $V$. Let $\mathcal{D}$ denote the differential operators with constant coefficients and $\mathcal{D}^{G}$ be the $G$-invariant operators. A ...
• 105
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### What is the status of this fifty-year-old conjecture of Kostant?

On page 3.27 of his 1963 thesis on the cohomology of homogeneous spaces as approached through the Eilenberg–Moore spectral sequence, Paul Baum states the following conjecture, which he attributes to B....
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### Normalizer of a reflection supgroup of a finite complex reflection group

Let $W \leq \operatorname{GL}(\mathbb{C}^\ell)$ be a finite complex reflection group acting on the finite dimensional complex vectorspace $\mathbb{C}^\ell$. Let $W' \leq W$ be a reflection subgroup, i....
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### coxeter element of a reflection group (reference request)

I am reading reflection groups and coxeter groups book by Humphreys. now I want to learn more about "coxeter element" of a reflection group. Can any body suggests me some good books to read more about ...
• 1,219
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### A result from Peter McMullen's thesis

The classical definition of regular polytopes is recursive. It says that a polytope is regular if its facets and vertex figures (both smaller-dimensional polytopes) are regular. The modern definition ...
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Let $G$ be a finite group acting on a complex vector space $V$ by pseudoreflections (i.e. every element of $G$ is a product of elements which fix hyperplanes in $V$). I would like to understand the ...