  The descent algebra $\Sigma(W)$ is a subalgebra of the group algebra $\Q W$
of a finite Coxeter group $W$, which supports a homomorphism with nilpotent
kernel and commutative image in the character ring of $W$. Thus $\Sigma(W)$ is
a basic algebra, and as such it has a presentation as a quiver with relations.
Here we construct $\Sigma(W)$ as a quotient of a subalgebra of the path algebra
of the Hasse diagram of the Boolean lattice of all subsets of $S$, the set of
simple reflections in $W$. From this construction we obtain some general
information about the quiver of $\Sigma(W)$ and an algorithm for the
construction of a quiver presentation for the descent algebra $\Sigma(W)$ of
any given finite Coxeter group $W$.
