A symmetry of the descent algebra of a finite Coxeter group.
Let \((W,S)\) be a finite Coxeter system. For each \(I\subset S\), let \(W_I\) be the parabolic subgroup of \(W\) generated by \(I\), then \(W^I=\{x\in W\mid l(x)<l(xs)\) for all \(s\in I\}\) is a cross-section of the left cosets of \(W_I\) in \(W\), consisting of the unique representatives of minimal length. The linear span \(\mathcal D_W\) of the elements \(X^I=\sum_{x\in W^I}x\) (\(I\subset S\)) is a subalgebra of the integral group algebra \(\mathbb{Z} W\), the descent algebra of \(W\). Furthermore, if \(\varphi_I\) denotes the character of \(W\) induced by the principal character of \(W_I\), then the linear map \(\theta\) defined by \(\theta(X_I)=\varphi_I\) for all \(I\subset S\) is a homomorphism of rings mapping the descent algebra into the ring of class functions of \(W\). For \(W\) of type \(A\), \textit{A.~Jöllenbeck} and \textit{C. Reutenauer} derived the identity \(\theta(X)(Y)=\theta(Y)(X)\) for all \(X,Y\in\mathcal D_W\) [in Abh. Math. Semin. Univ. Hamb. 71, 105-111 (2001; Zbl 0996.05118)], where class functions of \(W\) have been extended to the group algebra of \(W\) linearly. They conjectured that this symmetry property of \(\mathcal D_W\) holds for arbitrary finite Coxeter groups \(W\). In the paper under review the authors prove that this conjecture on this symmetry property of \(\mathcal D_W\) holds for arbitrary finite Coxeter groups \(W\). As a consequence, several properties of the characters of \(W\) afforded by the primitive idempotents of \(\mathcal D_W\) are derived by the authors at once, including a symmetry of the corresponding character table, and a combinatorial description of their intertwining numbers with the descent characters of \(W\).
scientific article; zbMATH DE number 2174976
scientific article