On the descending Loewy series of Solomon's descent algebra (Q1911781)

Let \(K\) be a field of characteristic 0 and \(n\) be a positive integer. For every element \(\sigma\) of the symmetric group \(S_n\) we put \(D(\sigma):=\{k\mid 1\leq k<n,\;k\sigma>(k+1)\sigma\}\). Let \(\Delta_n\) be the \(k\)-subspace of the group algebra \(KS_n\) generated by all elements \(\delta_D:=\sum_{\sigma\in S_n\atop D(\sigma)=D}\sigma\) where \(D\subseteq\{1,\dots,n-1\}\). Then \(\Delta_n\) is a subalgebra \(KS_n\) called Solomon's descent algebra. The Jacobson radical of \(\Delta_n\) is denoted by \(\text{Rad }\Delta_n\). In this paper, a description of the descending Loewy series \(((\text{Rad }\Delta_n)^j)_{j\geq 0}\) of \(\Delta_n\) is obtained.