Götz Pfeiffer,
# A Quiver Presentation for Solomon's Descent Algebra.

45 pages.
Preprint IRL-GLWY-2007-001.
arXiv:0709.3914v1 [math.RT].

## Abstract.

The descent algebra $\Sigma (W)$ is a subalgebra of the group algebra
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 $W$.
Available as DVI file (265 kB),
as pdf file (481 kB)
and as
compressed PostScript (283 kB) file.