Questions tagged [reflection-groups]

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6 votes
1 answer
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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 ...
2 votes
0 answers
219 views

Proving that a product of reflections and an orthogonal matrix is in $\mathrm{SO}_*(V)$

Let $V=(V,b)$ be a finite-dimensional vector space equipped with $b$ a symmetric and positive definite bilinear form. And let $\{e_1,\dotsc,e_n\}$ be a orthonormal basis for the subspace $\ker((P_A)^t)...
4 votes
0 answers
110 views

Polynomial invariants of infinite reflection groups

It is a famous theorem of Shepard-Todd that the ring of invariant polynomials of a finite complex reflection group $W$ acting on a complex vector space $V$ is actually itself a polynomial ring. In ...
1 vote
0 answers
117 views

Relation between C-groups and reflection groups

Take a set of reflections $\{r_1,\ldots,r_k\}$ of $\mathbb R^n$. Sometimes, the group presentation will turn out to be a C-group – this is where the regular planar polytopes in Euclidean space, ...
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1 vote
0 answers
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Numbers of reflections on a coset in reflection group

Consider $G_{k}$ a group of type Shephard–Todd. Given a reflection $\sigma \in G_{k}$, take $H$ the subgroup generated by $\sigma$. How to determine the number of reflections in the coset $\tau \cdot ...
3 votes
1 answer
116 views

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 ...
7 votes
0 answers
189 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 ...
2 votes
1 answer
123 views

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 $v\in V\backslash\{0\}$ (...
8 votes
1 answer
329 views

Which reflection groups can be enlarged?

Based on this question (which focuses on the case $E_8$) I wonder the following: Question: For each finite reflection group $\Gamma\subseteq\mathrm O(\Bbb R^d)$, what is the largest finite group $\...
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7 votes
1 answer
184 views

When are indiscrete reflection groups Coxeter groups?

A well-known theorem of Coxeter states that any discrete group $W$ which is generated by reflections across (possibly affine) hyperplanes in Euclidean space is a Coxeter group: it has a presentation ...
0 votes
0 answers
133 views

Counting special paths on a certain rectangle integer grid (binary matrix)

Crossposting from MSE after getting no answers. The bounty on the MSE question is still open, but not for long. Be advised that the comments of the MSE question regard an obsolete version, and that ...
5 votes
0 answers
189 views

Pseudoreflection groups in affine varieties

Suppose $\mathsf{k}$ is an algebraically closed field of zero characteristic. Chevalley-Shephard-Todd (C-S-T) Theorem in one of its equivalent versions is the following result: (C-S-T): Let $G$ be a ...
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5 votes
1 answer
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Finite group ${\rm Sp}_4({\Bbb F}_3)$: involutions coming from a 4-dimensional complex representation

I am interested in the finite unitary reflection group $G= G_{32}$, the group No. 32 in Table VII on page 301 of the paper: Shephard, G. C.; Todd, J., A. Finite unitary reflection groups. Canad. J. ...
10 votes
0 answers
293 views

Recognizing reflection subgroups of Coxeter groups

Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $...
19 votes
0 answers
329 views

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 ...
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10 votes
1 answer
269 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 ...
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2 votes
0 answers
151 views

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 ...
9 votes
1 answer
344 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 ...
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3 votes
1 answer
237 views

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 ...
10 votes
0 answers
308 views

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. ...
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2 votes
0 answers
184 views

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 $M(n)$...
4 votes
1 answer
266 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 $W$....
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13 votes
2 answers
619 views

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: ...
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1 vote
1 answer
206 views

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 ...
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3 votes
2 answers
383 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 ...
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12 votes
2 answers
594 views

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 ...
3 votes
0 answers
214 views

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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9 votes
1 answer
391 views

Bruhat order of reflection subgroups

Let $(W,S)$ be a Coxeter group, $T=\bigcup_{w\in W}wSw^{-1}$ its set of reflections, and $A\subseteq T$. From results of Dyer and Deodhar, we know that the subgroup $W_A$ generated by the elements of $...
2 votes
0 answers
288 views

the root lattice, reflections, and a coxeter element

Question: Is is possible to realise the positive root lattice $\Phi_{\Delta}^{>0}$ (viewed as an abstract poset) of a root system $\Phi_\Delta$ associated to a Dynkin or affine Dynkin diagram $\...
6 votes
1 answer
299 views

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 ...
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7 votes
0 answers
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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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5 votes
0 answers
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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....
0 votes
1 answer
290 views

SO(3) transformation that produces a reflection [closed]

This came up doing some research in quantum information. Let us consider two orthogonal three-dimensional unit vectors $v$ and $w$ $v^T\cdot w=0$, and the Householder transformation $H=I_{3}-2v\...
6 votes
2 answers
406 views

Centralizers of reflections in special 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 $m_{i,j}=m_{j,i}$...
50 votes
3 answers
3k views

The view from inside of a mirrored tetrahedron

Suppose you were standing inside a regular tetrahedron $T$ whose internal face surfaces were perfect mirrors. Let's assume $T$'s height is $3{\times}$ yours, so that your eye is roughly at the ...
7 votes
1 answer
321 views

Origin of the numbers game

The numbers game is a (one-player) game played on a finite graph with an initial assignment of numbers to its vertices, studied by Alon, Bj\"orner, Brenti, Donnelly, Eriksson, Krasikov, Mozes, Peres, ...
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2 votes
1 answer
173 views

Is inner product preserved only by the stabiliser in a finite reflection group?

Is the following statement true for finite reflection groups? Let $G$ be a finite reflection group acting on $\mathbb{R}^n$, let $x, y\in \mathbb{R}^n$ and let $z$ be in the orbit of $y$. If $\...
2 votes
0 answers
143 views

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 ...
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7 votes
2 answers
574 views

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 ...
9 votes
2 answers
458 views

Can the difference of non-conjugate pseudoreflections lie in the commutator subgroup?

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 ...