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We make some conjectures about tensor products of modular representations for a finite p-group in characteristic p, for p = 2 and p = 3 . The easiest to state is that if M is a module of odd dimension for a 2-group over an algebraically closed field of characteristic two, then k is the only direct summand of M ⊗ M ⁎ of odd dimension, and the remaining indecomposable summands have dimension divisible by four. As a consequence, the odd dimensional modules form an abelian group, where the composition law is to take the unique indecomposable summand of odd dimension of the tensor product, and where the inverse of an odd dimensional module is its dual. In characteristic three the conjectures are more complicated, and in larger characteristics it is not at all clear what the corresponding conjectures should be. We prove a number of theorems relating to these conjectures, and describe examples illustrating them. The examples were computed using the computer algebra package Magma . |
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