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

초록

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 F(p). This paper concerns the conjecture on the lambda-ring structure of a certain quotient ring RS(p(k))/I(p)k of RS(p(k)) when k >= 2, which was originally due to Kouwenhoven. To be more precise, he conjectured that the ideal I(p)k is closed under the exterior powers and RS(p(k))/I(p)k is equipped with the lambda-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(p(k))/I(p)k cannot have the lambda-ring structure for the induced exterior powers. (C) 2011 Elsevier Inc. All rights reserved.

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