Question

Texts: Consider the action of Gl₃(C) on Sl₃(C) (the vector space of 3x3 traceless matrices) by conjugation. Show that this representation is irreducible. i) I know that it has the weights (1, 0, -1), (0, 1, -1), (1, -1, 0), (0, 0, 0), (0, -1, 1), (-1, 1, 0), and (-1, 0, 1). ii) I also know that in general, if for each weight vector v, one can obtain the whole representation of a group G by the linear span of g•v, for all g in G, then our representation is irreducible. But I do not know how to use ii) in this case to prove that the representation above (g•A•g⁻¹) is irreducible. Thank you for your help.

          Texts: Consider the action of Gl₃(C) on Sl₃(C) (the vector space of 3x3 traceless matrices) by conjugation.
Show that this representation is irreducible.
i) I know that it has the weights (1, 0, -1), (0, 1, -1), (1, -1, 0), (0, 0, 0), (0, -1, 1), (-1, 1, 0), and (-1, 0, 1).
ii) I also know that in general, if for each weight vector v, one can obtain the whole representation of a group G by the linear span of g•v, for all g in G, then our representation is irreducible.
But I do not know how to use ii) in this case to prove that the representation above (g•A•g⁻¹) is irreducible.
Thank you for your help.

Added by Ronald W.

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