A generating set of Solomon's descent algebra
Let \(n\) be a positive integer. For any permutation \(\pi\) in the symmetric group \(S_n\), we denote by \(D(\pi)=\{i\mid 1\leq i\leq n-1,\;i\pi>(i+1)\pi\}\) the descent set of \(\pi\). Solomon (1976) discovered that the linear span \({\mathcal D}_n\) of the elements \(\Delta^D=\sum_{D(\pi)=D}\pi\), \(D\subseteq\{1,\dots,n-1\}\), is a subalgebra of the group ring \(KS_n\). The main result of this paper is: Theorem. For all \(n\), Solomon's descent algebra \({\mathcal D}_n\) is generated by elements \(\Delta^{\{1,\dots,k\}}\), \((n-1)/2\leq k\leq n-1\). The ring \(Cl_K(S_n)\) of class functions of \(S_n\) is generated by (1) the irreducible \(S_n\)-characters \(\chi^p\) corresponding to the hook partitions \(p=(n-k).1^k\) of \(n\) such that \(k\geq(n-1)/2\) (Corollary 1); (2) the \(S_n\)-characters \(\chi_i\) (\(0\leq i\leq n-1\)) corresponding to the natural action of \(S_n\) on the \(i\)-th homogeneous component of the quotient ring \(\mathbb{Q}[x_1,\dots,x_n]/I\), where \(I\) is the ideal of the polynomial ring \(\mathbb{Q}[x_1,\dots,x_n]\) generated by the symmetric polynomials without constant terms (Corollary 3).
scientific article; zbMATH DE number 1924686