The direct sum of complex representation rings $R_*\oplus R\Sigma_n$, for $\Sigma_n$ the $n$th symmetric group is also the free $\lambda$-ring on one generator. Here, we take a product obtained from induction on representations. But, each $R\Sigma_n$ also has an internal tensor product, making $R_*$ a hopf ring. Does anyone know, or know of a reference that explains, how the $\lambda$-ring operations relate to the internal tensor product?
There is also a nice little book by Hoffman, called "tau-rings and wreath-product representations", Lecture Notes in Mathematics 746