On the irreducible matrix representations of finite \(p\)-groups over local factorial rings (Q2730542)

The authors study conditions under which any irreducible matrix representation of a finite \(p\)-group over some local factorial ring \(K\) of characteristic zero is irreducible also over the quotient field of the ring \(K\). The following result is proved. Let \(G\) be a finite \(p\)-group of order \(|G|>1\) (\(p\neq 2\)) and let \(K\) be a local factorial ring of characteristic zero with field of residues of characteristic \(p\) and \(\varepsilon\in K\) (\(\varepsilon^p=1\), \(\varepsilon\neq 1\)). The group \(G\) has no irreducible matrix \(K\)-representation which is reducible over the quotient field of the ring \(K\) if and only if \(K\) is a discretely normed ring.