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

Hello pricey customer to our community We will proffer you an answer to this query gr.group principle – Arbitrarily sizable finite irreducible matrix teams in queer dimension? ,and the respond will breathe typical by way of documented data sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

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 $$B_d$$.

we are going to proffer you the answer to gr.group principle – Arbitrarily sizable finite irreducible matrix teams in queer dimension? query through our community which brings all of the solutions from a number of dependable sources.