Formulae for symmetric powers and tensor products of modular representations of elementary abelian p-groups

In the landscape of modular representation theory, the study of decomposition formulas for symmetric powers and tensor products in the context of elementary abelian p-groups remains underdeveloped in the existing literature. This research addresses this gap by systematically developing explicit mathematical formulas for these operations within such groups. A central objective is to extend the results on indecomposable representations of cyclic groups to specific classes of representations for elementary abelian p-groups, with a focus on deriving decomposition formulas for symmetric powers and tensor products into direct sums.

Let V2 denote the two-dimensional faithful indecomposable module for elementary abelian p-groups. We define Vi as the dual module Si−1(V2)∗, where S represents sym-metric powers, and ∗ denotes the dual (or contragredient) module.

Our approach stands out due to the special focus we give to delving into the details of computations and the methodical examination of formulaic expressions.
We utilized the Magma calculator within the Magma computational algebra software as our primary methodology. Through the use of Magma’s calculator, we efficiently com-puted the tensor products and symmetric powers of modular representations of elementary abelian p-groups, providing precise outcomes in our research.

In this research, we analyze the structure of the tensor products of indecomposable modules over an elementary abelian p-group G of order q = pn, where p is a prime and k is a field of characteristic p.
One of the core results is a new decomposition theorem.
For all i < q, and for any integer i not divisible by p, the tensor product V2 ⊗ Vi decom-poses as Vi+1 ⊕ Vi−1.
In particular, we prove that the tensor product V2 ⊗Vp is indecomposable, provided p ̸= q. Furthermore, we extend existing theorems on symmetric powers by establishing their rela-tionship with the Heller (shift) operator. We also derive general criteria for decomposing the tensor products Vi ⊗ Vj into direct sums based on the indices i and j, and the param-eters p and q, providing foundational tools to verify and generalize the main theorem.

Finally, drawing on our results, we propose a conjecture that identifies a decomposition pattern for the tensor products of modules Vi and Vj . These advances bridge theoretical and computational gaps in modular representation theory, offering both structural insights and practical methodologies for further exploration.