Cyclic Eulerian Elements

LetSnbe the symmetric group on {1,…,n} and Q[Sn] its group algebra over the rational field; we assumen ≥ 2. π ∈ Snis said a descent ini, 1 ≤ i ≤ n - 1, if π(i) > π (i + 1); moreover, π is said to have a cyclic descent if π(n) > π(1). We define the cyclic Eulerian elements as the sums of all elements inSnhaving a fixed global number of descents, possibly including the cyclic one. We show that the cyclic Eulerian elements linearly span a commutative semisimple algebra of Q[Sn], which is naturally isomorphic to the algebra of the classical Eulerian elements. Moreover, we give a complete set of orthogonal idempotents for such algebra, which are strictly related to the usual Eulerian idempotents.