Some formulae relating modular representations of elementary abelian p-groups

Let p > 0 be a prime, k a field of characteristic p and G an elementary abelian p-group of order q = p^n. Let W be an indecomposable kG-module of dimension 2 and define V_i =S^{i−1}(W)^∗ for each i = 1, . . . , q. We show that V2 ⊗ Vi∼=Vi+1 ⊕ Vi−1 provided i is not divisible by p. Our results generalise results of Almkvist and Fossum for representations of cyclic groups of order p. We show how our results give formulae for the direct sum decomposition of Vi ⊗ Vj for all i < p and j < q modulo summands projective to M := V_p ⊕V_{2p} ⊕ ... ⊕ V_q, and conjecture that these formulae extend to the case i < q and j < q. We provide some evidence for our conjecture.

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