# gr.group principle – Arbitrarily sizable finite irreducible matrix teams in queer dimension? Answer

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gr.group principle – Arbitrarily sizable finite irreducible matrix teams in queer dimension?

I deem a finite irreducible matrix group $$Gammasubseteqmathrm{GL}(Bbb R^d)$$. I’m within the maximal dimension of $$Gamma$$ relying on $$d$$. But this query makes solely sense if there may be an higher restrict.

In plane dimension there isn’t a such restrict. This is best seen in dimension $$d=2$$, the place we have now the cyclic teams or dihedral teams of arbitrarily sizable dimension. More typically, in dimension $$d=2n$$ we will select the balance group of the $$n$$-th cartesian energy of a daily $$ok$$-gon:

$$P=overbrace{C_ktimes cdotstimes C_k}^{textual content{n occasions}}.$$

This group is irreducible and will get arbitrarily sizable with $$ktoinfty$$.

Question: What about queer dimensions? Can there breathe arbitrarily sizable finite irreducible matrix teams in dimension $$d=2n+1$$?

For instance, in dimension $$d=3$$ we have now the arbitrarily sizable balance teams of prisms and antiprisms, that are reducible. The largest irreducible group might be the balance group of the icosahedron.

I’ve the emotion that in sufficiently sizable queer dimensions, the most important such group is the reflection group $$B_d$$.

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