Questions tagged [reflection-groups]

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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 ...
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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 ...
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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. ...
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 $M(n)$...
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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 $m_{i,j}=m_{j,i}$...
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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 ...