# $\lambda$-rings and hopf-rings

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?

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I thought Hopf means having a product and a coproduct. You choose to dualise one of your products? –  Vladimir Dotsenko Mar 7 '12 at 18:59
@VladimirDotsenko: Having a product and coproduct makes it a hopf algebra. A hopf ring has an additional product that satisfies some further axioms with the coproduct and original product. –  Joe Johnson Mar 7 '12 at 19:27
Reference for the first statement: the book "$\lambda$-rings and the representation theory of the symmetric group", by D. Knutson. –  Allen Knutson Mar 8 '12 at 3:40
Thanks, I knew that and should have remembered it. It's just one of those terms that always confuses me (algebras have more structures than rings, but Hopf rings have more structures than Hopf algebras). Did you trace through Macdonald's "Symmetric functions and Hall polynomials"? I think something of what you need must be contained in the relevant chapter there. –  Vladimir Dotsenko Mar 8 '12 at 5:43

## 1 Answer

There is also a nice little book by Hoffman, called "tau-rings and wreath-product representations", Lecture Notes in Mathematics 746

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Hoffman's book is very nice indeed. –  Joe Johnson Feb 11 '13 at 15:53