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

Ram Karan Choudhary, Sunil Kumar Prajapati

Abstract

In this article, we present a combinatorial formula for computing the Wedderburn decomposition of the rational group algebra associated with an ordinary metacyclic $p$-group $G$, where $p$ is any prime. We also provide a formula for counting irreducible rational representations of $G$ with distinct degrees and derive a method to explicitly obtain all inequivalent irreducible rational matrix representations of $G$.

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

Abstract

In this article, we present a combinatorial formula for computing the Wedderburn decomposition of the rational group algebra associated with an ordinary metacyclic -group , where is any prime. We also provide a formula for counting irreducible rational representations of with distinct degrees and derive a method to explicitly obtain all inequivalent irreducible rational matrix representations of .

Paper Structure

This paper contains 8 sections, 21 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Complex representations of ordinary metacyclic $p$-groups
  4. Rational representations of ordinary metacyclic $p$-groups
  5. Rational matrix representations of ordinary metacyclic $p$-groups
  6. Rational group algebra of ordinary metacyclic $p$-groups
  7. Examples
  8. Acknowledgements

Key Result

Theorem 1

Let $p$ be a prime and $\zeta_d$ a primitive $d$-th root of unity for some positive integer $d$. Consider an ordinary, finite, non-cyclic metacyclic $p$-group $G$, with a unique reduced presentation: for certain integers $m, n, r$ and $s$ (as defined in prest:metacyclic). Then we have the following.

Theorems & Definitions (43)

  • Theorem 1
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • Remark 5
  • Proposition 6
  • proof
  • ...and 33 more