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My problem is the following:

enter image description here

I have difficulties in answering questions (c), (d) and (e).

For (c) my answer was $\sqrt{x^{2}+t^{2}}$ and yes, the group forms the group of all isometries since the group contains those matrices that translate and rotate the 2D space.

For (d) I have said that there is no difference since E(2) is the symmetry group of 2D space.

For (e) I have tried using the theorem which states that if D is a irrep of a group G and there exists a matrix B which commutes with B then $B=\lambda 1$. Thus, I tried forming a 3x3 matrix and imposing conditions but that has only led me to the diagonal elements being equal and the elements below the main diagonal being zero. Is there some other theorem that I could apply?

Is this the way to approach the questions?

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  • $\begingroup$ Are you sure that Galilean boosts preserve the line element $\mathrm{d}x^2 + \mathrm{d}t^2$? $\endgroup$ – gj255 Apr 10 '17 at 16:14
  • $\begingroup$ it certainly preserves the length $dx^2$ and the time interval $dt^2$ separately by ordinary geometry and Newtonian definition of time interval. $\endgroup$ – ZeroTheHero Apr 10 '17 at 21:47
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  • Part c) Surely the length is $x^2+t^2$.
  • Part d) I don't think your matrix representation for $T(a,b,v)$ is $E(2)$. See in particular Eq.(1) of this paper.
  • Part e) Galilei in (1+1)
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  • $\begingroup$ I will have a look at the paper. Meanwhile, should I understand from your answer for e) that since the dimension of the space is 2 while the matrix is 3x3 then I should be able to reduce it to block that is 2x2 and another block that is 1x1? $\endgroup$ – Tarabostes Delectus Apr 10 '17 at 13:51
  • $\begingroup$ You may not be able to reduce it: the representation may be indecomposable (rather than irreducible). However the representation theory of this type of group is rather delicate. I don't think you can bring it to full block diagonal form, at least not using real transformations. (I could be wrong but it looks improbable.) $\endgroup$ – ZeroTheHero Apr 10 '17 at 14:00

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