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A quiver presentation for Solomon’s descent algebra. (English) Zbl 1203.20004
Summary: The descent algebra $$\Sigma(W)$$ is a subalgebra of the group algebra $$\mathbb{Q} W$$ 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 $$W$$.

##### MSC:
 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16G20 Representations of quivers and partially ordered sets 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05E10 Combinatorial aspects of representation theory 05E15 Combinatorial aspects of groups and algebras (MSC2010)
##### Software:
CHEVIE; GAP; OEIS; ZigZag
Full Text:
##### References:
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