Study of Matrix Units of the Group algebras 𝐾𝐺𝑟and 𝐾𝑆𝐺𝑟

Objectives: The purpose of this work is to study 𝑝-Bratteli diagrams arising from a class of group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 for every odd prime 𝑝, to define the irreducible representations using mutually orthogonal primitive idempotents and to index the complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Method: We use primitive idempotents to define irreducible representations and systematically index the irreducible representations to establish vertices for the 𝑝-Bratteli diagram. Also, we compute the matrix units using the orthogonal primitive idempotents of the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟. Findings: The complete set of irreducible representations for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 has been successfully indexed, and matrix units for the group algebras 𝐾𝐺𝑟 and 𝐾𝑆𝐺𝑟 were determined using the orthogonal primitive idempotents. Novelty: This work provides a structured approach to constructing 𝑝-Bratteli diagrams for the group algebras related to odd primes, contributing new insights into their irreducible representations and matrix unit’s structures. Keywords: Primitive idempotent; Irreducible representation; Matrix units; Young diagram; Bratteli diagram