Rational representations and rational group algebra of VZ \(p\) -groups (Q6668355)

Let \(G\) be a finite group, \(\mathrm{Irr}(G)\) the set of all complex irreducible characters of \(G\) and \(p\) a prime number. If \(\chi \in \mathrm{Irr}(G)\), let \(\Omega(\chi)=m_{\mathbb{Q}}(\chi) \sum_{\sigma \in \mathcal{G}}\chi^{\sigma}\), where \(\mathcal{G}=\mathrm{Gal}(\mathbb{Q}(\chi)/\mathbb{Q})\) and \(m_{\mathbb{Q}(\chi)}\) is the Schur index of \(\chi\) over \(\mathbb{Q}\). Then \(\Omega(\chi)\) corresponds to the character of an irreducible \(\mathbb{Q}\)-representation \(\rho\) of \(G\) and, conversely, if \(\rho\) is an irreducible \(\mathbb{Q}\)-representation of \(G\), then there exists \(\chi \in \mathrm{Irr}(G)\) such that \(\Omega(\chi)=\rho\).\N\NIn this paper, the authors study irreducible rational matrix representations for some classes of \(p\)-groups. For any \(p\)-group \(G\), they present two algorithms for computing an irreducible rational matrix representation of \(G\) that affords the character \(\Omega(\chi)\).\N\NA group \(G\) is called a VZ-group if all its nonlinear irreducible characters vanish off the center. Using their results on VZ \(p\)-groups, the authors explicitly determine all inequivalent irreducible rational matrix representations of all \(p\)-groups of order \(\leq p^{4}\). At the same time, they investigate the Wedderburn decomposition of \(\mathbb{Q}G\), with a specific focus on VZ \(p\)-groups.