TITLE: Grothendieck group of a symmetric monoidal category is a lambda ring?
QUESTION [7 upvotes]: I understand that taking the Grothendieck group of a braided monoidal (abelian) category gives us a commutative ring and that taking that of a symmetric monoidal (abelian) category gives us a $\lambda$-ring.  Now, I have simply seen this (latter) fact stated on the internet by reputable mathematicians, but I have not been able to find a reference explaining how one gets all that extra structure ($\lambda$-rings seem quite complicated!) from one extra property that symmetric monoidal categories have.
I was wondering if anybody could point me in the right direction or shed some light on the details?
Thanks!

REPLY [8 votes]: The $\lambda$-structure is given by taking exterior powers. This is the main motivation I know for defining $\lambda$-rings in the first place. (You need an action of $S_n$ on an $n^{th}$ tensor power $V^{\otimes n}$ to define the exterior power, which is what being symmetric monoidal gets you; in the braided monoidal case you only get an action of $B_n$.)