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Let $G$ be a reductive group over $\mathbb{C}$ and $Rep(G)$ the category of rational representations.
Is there a "nice" (let's say combinatorical) description of the generators of $Rep(G)$ as a tensor category?

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    $\begingroup$ You can work over any algebraically closed field of characteristic 0 if you want (this ensures complete reducibiity of representations). If $G$ is semisimple and simply connected, the finite dimensional irreducible representations with a fundamental highest weight may suffice for your purpose. Removing the simply connected condition complicates the picture somewhat. Adding a central torus is a further complication, even though its irreducible rational representations are just characters. So it's useful to make your assumptions as precise as possible. $\endgroup$
    – Jim Humphreys
    Commented May 14, 2013 at 13:48
  • $\begingroup$ Thanks for the nice answer. Semisimple and simply connected is a start. Do you know any reference on this? Anyway i would be really interested why the situation in the non simply connected case is harder. Again, i would appreciate any reference. Best, Oliver. $\endgroup$
    – Oliver Straser
    Commented May 14, 2013 at 14:59
  • $\begingroup$ I'm not sure what you mean by "generators of $Rep(G)$ as a tensor category": do you allow the operation of taking sub-representation in the way you generate? Do you want generators for the objects of $Rep(G)$, and generators for the morphisms of $Rep(G)$? $\endgroup$
    – André Henriques
    Commented May 14, 2013 at 18:53
  • $\begingroup$ Yes taking sub-representations would be fine. Also it would be enough to have generators for the objects. $\endgroup$
    – Oliver Straser
    Commented May 14, 2013 at 19:26
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    $\begingroup$ @O.: Irreps are parametrized by dominant integral highest weights, formalized in the representation ring (Grothendieck ring). For a simply connected semisimple group (or compact Lie group), it's equivalent to work with the Lie algebra; e.g. Brocker & tom Dieck *Representations of Compact Lie Groups, II.7. or Bourbaki Lie Groups and Lie Algebras, Chap. 8, section 7 (and exercise 27). Fundamental dominant weights yield standard "generators". For a not simply connected group, limit to dominant weights in cosets of the weight lattice mod root lattice. (No easy "generators".) $\endgroup$
    – Jim Humphreys
    Commented May 17, 2013 at 13:27

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