A Combinatorial Technique for the Wedderburn Decomposition of Rational Group Algebras of Nested GVZ p-groups

In this article, we present a combinatorial formula for the Wedderburn decomposition of rational group algebras of nested GVZ p-groups, where p is an odd prime. Using this formula, we derive an explicit combinatorial expression for the Wedderburn decomposition of rational group algebras of all two-generator p-groups of class 2. Additionally, we provide explicit combinatorial formulas for the Wedderburn decomposition of rational group algebras of certain families of nested GVZ p-groups with arbitrarily large nilpotency class. We also classify all nested GVZ p-groups of order at most p⁵ and compute the Wedderburn decomposition of their rational group algebras. Finally, we determine a complete set of primitive central idempotents for the rational group algebras of nested GVZ p-groups.