Let $G$ be a compact Lie group and let $R(G)$ denote its complex representation ring. If $G$ is simply connected, such as $G\_2$, $F\_4$ or $E\_8$, then it is known that $R(G)$ is a polynomial ring [F. Adams, *Lectures on Lie groups* ('69), Theorem 6.41].


**Question:** Are the representation rings of the adjoint groups $E\_6$ and $E\_7$ polynomial rings as well?


(Edit: the following articles are irrelevant as they consider the simply connected cover rather than the adjoint group.) In [T. Watanabe, *The Chern Character of the Symmetric Space EI*, Publ. RIMS ('95)] it is claimed that it is shown in [H. Minami, *K-Groups of Symmetric Spaces I+II*, Osaka J. Math. (75'+76')] that $R(E\_6)$ is a polynomial ring. I wasn't able to locate this precise claim in the second source, though.
I've found no reference concerning $R(E\_7)$.



Representation rings of exceptional Lie groups
As Victor points out in his comment, your formulation is somewhat out of focus. The general fact is that the representation ring of a connected semisimple compact Lie group is a polynomial algebra iff each simple factor of the group is simply connected or is an odd dimensional special orthogonal group. It's useful to consult the treatment in the Springer GTM 98 *Representations of Compact Lie Groups* by Brocker and tom Dieck (pages 255-256) along with the references given there. I've checked for example the useful paper by Steinberg referred to there.


So for exceptional groups of type $E\_6, E\_7$ which are of adjoint rather than simply connected type, the representation ring fails to be a polynomial ring (but is always finite generated).


P.S. Concerning notation, you should write $R(G)$ for the representation ring of a given compact Lie group, since this can vary with the isogeny type in a fixed Lie type such as $E\_6, E\_7$. The general result found in references like those I mentioned is that $R(G)$ is isomorphic to the ring of invariants of the Weyl group $W$ of a maximal torus $T$ in the integral group ring of the character group $X(T)$. For an adjoint group $G$ this is the *root lattice* rather than the full weight lattice of the root system, which can complicate the description a lot. I'm not aware of explicit formulas in the literature for your special cases, but classical types appear in some Bourbaki exercises (see Chap. VI, $\S3$ and Chap. IX, $\S7$ of *Lie Groups and Lie Algebras*). 


One small example: In type $A\_1$, take $G= \mathrm{SU}(2)$ (simply connected) with $X(T)$ spanned by fundamental weight $\varpi$ and $W$ of order 2. Formally setting $x = e^\varpi$, we get $\mathbb{Z}[X(T)] = \mathbb{Z}[x,x^{-1}]$ (Laurent polynomials) with a reflection in $W$ sending $x \mapsto x^{-1}$; so $R(G) \cong \mathbb{Z}[x+x^{-1}]$.Taking $G= \mathrm{SO}(3)$ (adjoint), the root `$\alpha = 2\varpi$ spans $X(T)$, so instead $R(G) \cong \mathbb{Z}[x^2 + x^{-2}]$. (Since $A\_1 = B\_1$, the ring is polynomial in both cases.) 


