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A Combinatorial Formula for the Wedderburn Decomposition of Rational Group Algebras of Split Metacyclic $p$-groups

Ram Karan Choudhary, Sunil Kumar Prajapati

Abstract

In this article, we present a concise combinatorial formula for efficiently determining the Wedderburn decomposition of rational group algebra associated with a split metacyclic $p$-group $G$, where $p$ is an odd prime. We also provide a combinatorial formula to count irreducible rational representations of $G$ of distinct degrees.

A Combinatorial Formula for the Wedderburn Decomposition of Rational Group Algebras of Split Metacyclic $p$-groups

Abstract

In this article, we present a concise combinatorial formula for efficiently determining the Wedderburn decomposition of rational group algebra associated with a split metacyclic -group , where is an odd prime. We also provide a combinatorial formula to count irreducible rational representations of of distinct degrees.
Paper Structure (8 sections, 10 theorems, 21 equations)

This paper contains 8 sections, 10 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Review of rational representations of $C_{p^n} \times C_{p^m}$
  3. Wigner-Mackey method
  4. Complex representations of split metacyclic $p$-groups
  5. Rational representations of split metacyclic $p$-groups
  6. Rational group algebra of split metacyclic $p$-groups
  7. Examples
  8. Acknowledgements

Key Result

Theorem 1

Let $p$ be an odd prime and let $\zeta_d$ be a primitive $d$-th root of unity. Consider a finite non-abelian split metacyclic $p$-group $G = \langle a, b\mid a^{p^n} = b^{p^m} = 1, bab^{-1} = a^r \rangle$, where $n\geq 2, m \geq 1$, $(r, p^n) = 1$ and $r$ has multiplicative order $p^s$ modulo $p^n$

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 2
  • Remark 3
  • Lemma 4
  • proof
  • Proposition 5
  • Theorem 6
  • Lemma 7
  • proof
  • Lemma 8
  • ...and 12 more