Homomorphisms between Solomon's descent algebras (Q1204547)

For a permutation \(\sigma = \sigma_ 1 \dots \sigma_ n\) let the descent set of \(\sigma\) be \(\text{Des}(\sigma) = \{i: 1 \leq i \leq n-1\) and \(\sigma_ i \geq \sigma _{i+1}\}\) and define the element \(D_{\subseteq S} = \sum_{\text{Des}(\sigma)}\sigma\), where \(S \subseteq \{1,\dots,n-1\}\). The linear span of these elements forms a \(2^{n-1}\)-dimensional subalgebra \(\Sigma_ n\) of the group algebra of the symmetric group \(\mathbb{Q}[S_ n]\), called Solomon's descent algebra. For any \(s\), \(0 < s < n\), the authors construct a surjective homomorphism \(\Delta_ s: \Sigma_ n \to \Sigma_{n-s}\) and an embedding \(\Gamma_ s: \Sigma_{n-s} \to \Sigma_ n\). The homomorphisms \(\Delta_ s\) are also defined as derivations of the free associative algebra \(\mathbb{Q}\langle t_ 1,t_ 2,\dots\rangle\) sending \(t_ i\) to \(t_{i-s}\) in case one identifies the basis element \(D_{\subseteq S}\) of \(\Sigma_ n\) with some word on the alphabet \(T = \{t_ 1,t_ 2,\dots\}\).