Some Formulae Relating Modular Representations of Elementary Abelian P-Groups

Let p>0 be a prime, a field of characteristic p and G an elementary abelian p-group of order q=p^n . Let W be an indecomposable G -module of dimension 2 and define V_i = S^i-1(W)^* for each i=1, … , q. We show that V_2 ⊗ V_i ≅ V_i+1⊕ V_i-1 provided i is not divisible by p. Our results generalise results of Almkvist and Fossum (1978) for representations of cyclic groups of order p. We show how our results give formulae for the direct sum decomposition of V_i ⊗ V_j for all i