Nilpotents of representation rings of finite p-groups

The representation ring a(G) of a finite group G provides a context in which to study the behavior of the module category under tensor product. Much work has been devoted to the semisimplicity question for representation rings, byt studying the existence and degree of nilpotents in a(G). In this paper, we construct a nilpotent of degree 3 in the representation ring a(Z/3 x Z/3), apparently the first explicit construction of such an element in odd characteristic. We make a number of observations on the general nilpotence question, and discuss applications of techniques developed in this paper to related questions.