A remark on a theorem of Ritter and Segal (Q2705006)

The paper concerns the rational character ring \(R_\mathbb{Q}(P)\) of a \(p\)-group \(P\), with \(p\) a prime number. Theorem: Any non-trivial irreducible \(\mathbb{Q}[P]\)-module \(V\) is of the form \(V=\mathbb{Q}(P/Q)-\mathbb{Q}(P/R)\), where \(R>_pQ\) are subgroups of \(P\) and where, for \(U\leq P\), \(\mathbb{Q}(P/U)\) denotes the \(\mathbb{Q}\)-span of the \(P\)-set \(P/U\). A consequence is \(\dim_\mathbb{Q} V\leq(p-1)[P:E]\) if \(E\) is an elementary Abelian subgroup of \(P\), or more generally, \(\dim_\mathbb{Q} V\equiv\dim_\mathbb{Q} V^{N_k}\bmod p^k(p-1)\) with \(N_k=\bigcap_{[P:U]=p^k}U\) and \(k\geq 0\). There is also a sort of converse of the statement in the theorem. The paper generalizes work of \textit{G. Segal} [Q. J. Math., Oxf. II. Ser. 23, 375-381 (1972; Zbl 0338.20017)] and the reviewer [J. Reine Angew. Math. 254, 133-151 (1972; Zbl 0242.20003)] on the surjectivity of the natural map from the Burnside ring of \(P\) to \(R_\mathbb{Q}(P)\).