λ-Ring structure of the Green ring of a cyclic p-group

摘要

Let F_p be a field of order p, G a cyclic group of order p~k, and RS(p~k) the Green ring of G over Fp. This paper concerns the conjecture on the λ-ring structure of a certain quotient ring RS(pk)/Ipk of RS(p~k) when k≥2, which was originally due to Kouwenhoven. To be more precise, he conjectured that the ideal Ipk is closed under the exterior powers and RS(pk)/Ipk is equipped with the λ-ring structure for the induced exterior powers. We show that Kouwenhoven's conjecture turns out to be true when p=2, but false when p=3. For other primes except p=2,3, it will be demonstrated that RS(pk)/Ipk cannot have the λ-ring structure for the induced exterior powers.