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Is there a simple characterisation of all rings which appear as $K_0$ of finite groups? By $K_0$ of a finite group $G$ I mean $K_0(\mathbb C[G])$ which in the same as a ring of virtual characters of finite G. So they must be finitely generated free as abelian groups and carry $\lambda$-structure. But what else?

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  • $\begingroup$ One more characterization is the fact that these are based rings: they have a basis with respect to which all the structure constants are positive integers. In addition, the dimension (and one can talk about the dimension here, using for example the theory of Frobenius Perron) of any simple element needs to be an integer and will divide the dimension of the ring (which can also be defined using FP). Of course, they are also commutative. I do not think, though, that these characterizations together with commutativity and the $\lambda$-ring structure is enough. $\endgroup$
    – Ehud Meir
    Commented Sep 1, 2015 at 16:09

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