Abstract 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 .

中文翻译：

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有限p群上模张量积的一些猜想及其后果

摘要 我们对特征为 p 的有限 p 群的模表示的张量积做了一些猜想，对于 p = 2 和 p = 3。最简单的表述是，如果 M 是特征为 2 的代数闭域上的 2 群的奇维模，则 k 是奇维 M ⊗ M ⁎ 的唯一直接被加数，其余不可分解的被加数有可被四整除的维数。因此，奇维模形成一个阿贝尔群，其中组合定律是取张量积的奇维的唯一不可分解和，而奇维模的逆是它的对偶。在特征三中，猜想更复杂，而在更大的特征中，对应的猜想应该是什么完全不清楚。我们证明了一些与这些猜想有关的定理，并描述了说明它们的例子。这些例子是使用计算机代数包 Magma 计算的。