For finite groups it is well-known that primitive central idempotents of the rational group algebra \(\mathbb{Q} G\) are of the form \(\tfrac{\chi(1)}{|G|}\sum_g\chi(g^{-1})g\), where \(\chi\) is an irreducible character. In this paper the authors provide a description of primitive central idempotents of the rational group algebra for finite nilpotent groups by means of the lattice of subgroups. For Abelian groups such a description is well-known [S. Perlis and G. L. Walker, Trans. Am. Math. Soc. 68, 420-426 (1950; Zbl 0038.17301)].
Let \(G\) be a finite nilpotent group. For a subset \(A\) of \(G\) let \(\widehat A=\tfrac 1{|A|}\sum_{a\in A}a\). If \(N\) is a normal subgroup of the group \(G\) then for \(N=G\) let \(\varepsilon(G,G)=\widehat G\), and for \(N\neq G\) let \(\varepsilon(G,N)=\prod(\widehat N-\widehat M)\), with \(M\) ranging over all subgroups of the group \(G\) containing \(N\) for which the factor group \(M/N\) is a minimal normal subgroup of the factor group \(G/N\). The primitive central idempotents of the rational group algebra of the group \(G\) are of the form \(\sum_g\varepsilon(G_m,H_m)^g\), where \(G_m\) and \(H_m\) are subgroups of the group \(G\) satisfying certain conditions.