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arXiv · 2401.07101

Rational group algebras of generalized strongly monomial groups: primitive idempotents and units

Gurmeet K. Bakshi·Jyoti Garg·Gabriela Olteanu
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Abstract

We present a method to explicitly compute a complete set of orthogonal primitive idempotents in a simple component with Schur index 1 of a rational group algebra $\mathbb{Q}G$ for $G$ a finite generalized strongly monomial group. For the same groups with no exceptional simple components in $\mathbb{Q}G$, we describe a subgroup of finite index in the group of units $\mathcal{U}(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$ that is generated by three nilpotent groups for which we give explicit description of their generators. We exemplify the theoretical constructions with a detailed concrete example to illustrate the theory. We also show that the Frobenius groups of odd order with a cyclic complement is a class of generalized strongly monomial groups where the theory developed in this paper is applicable.

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Gurmeet K. Bakshi, Jyoti Garg, Gabriela Olteanu. 2024-01-13. Rational group algebras of generalized strongly monomial groups: primitive idempotents and units. https://arxiv.org/abs/2401.07101

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