On the Quiver of the Descent Algebra

Using a result of T. P. Bidigare [Bidigare, 1997], we identity the descent algebra Σk(W ) (over a field k) of a finite Coxeter group W with a subalgebra of kF , an algebra built from the hyperplane arrangement associated to W . Specifically, Σk(W ) is anti-isomorphic to the W -invariant subalgebra (kF) . We use this identification and results about kF to study Σk(W ). We construct a complete system of primitive orthogonal idempotents for Σk(W ) and describe the simple and projective indecomposable Σk(W )-modules. The main result is an explicit construction of a W -equivariant surjection kQ ։ kF , where Q is the quiver of kF . Consequently, we obtain a surjection (kQ) ։ (kF) that can be used to gain information about the quiver of Σk(W ); as an application we derive the quiver of Σk(Sn).