Questions tagged [reflection-groups]

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0answers
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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 ...
3
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1answer
145 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 ...
9
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0answers
198 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. ...
2
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0answers
132 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
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1answer
160 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 $...
10
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2answers
332 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: ...
1
vote
1answer
122 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 ...
2
votes
2answers
259 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 ...
12
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2answers
455 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
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0answers
157 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 ...
9
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1answer
247 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
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0answers
257 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
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1answer
229 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 ...
7
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223 views

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....
5
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0answers
129 views

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
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1answer
148 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\...
4
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2answers
271 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
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3answers
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 ...
6
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1answer
271 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, ...
2
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1answer
168 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
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0answers
135 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 ...
6
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2answers
492 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
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2answers
405 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 ...