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Where can one find a table of generator–relator expressions for representation rings $R(G)$ of simple Lie groups $G$ and explicit maps between them? For example, given a maximal torus $T < G$ or a cover $\tilde G \to G$, I'd like this hypothetical table to have descriptions of the induced $$R(G) \rightarrowtail R(T),$$ $$R(G) \to R(\tilde G)$$ in terms of where they take generators. In principle, one could do this for oneself, but surely it's already written down somewhere.

edit: I'm aware that $R(T)$ is a Laurent polynomial ring, that $R(G) \cong R(T)^W$, and that $R(G)$ is a polynomial ring if $\pi_1 G$ is torsion-free or $G = \mathrm{SO}(2n+1)$, know what this implies for simply-connected classical groups, and know references to papers giving expressions for simply-connected exceptional groups. What I'm interested in is explicit descriptions for non–simply-connected groups in terms of their maximal tori and indications of where generators go under homomorphisms (including covering maps).

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  • $\begingroup$ May be I missed something, but isn't this equivalent to A) having character formulas for the irreducible reps of $G$, B) formulas for how a tensor product of two irreducible reps splits into a direct sum of irreducibles, and C) knowing which cosets of the root lattice are available as weights for non-simply-connected groups? $\endgroup$ – Jyrki Lahtonen Sep 13 '16 at 3:11
  • $\begingroup$ I guess it is the same as that, just more compact. Do you have a reference that has all that information? $\endgroup$ – jdc Sep 14 '16 at 20:22
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    $\begingroup$ I think "Representations of Compact Lie Groups" by tom Dieck and Bröcker has some examples. $\endgroup$ – Todd Leason Sep 17 '16 at 17:22
  • $\begingroup$ It definitely has some. I'd looked through it before asking the questions but I'm hoping for something more thorough. $\endgroup$ – jdc Sep 18 '16 at 4:04

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