The colored Eulerian descent algebra

Solomon's Descent Algebra Revisited

Starting from a non-standard definition, the descent algebra of the symmetric group is investigated. Homomorphisms into the tensor product of smaller descent algebras are defined. They are used to construct the irreducible representations and to obtain the nilpotency index of the radical.

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 sy...

On the Descent Algebra of Type D

Eulerian Circuits with No Monochromatic Transitions in Edge-colored Digraphs

Let G be an eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of G is an eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compa...

A Decomposition of Solomon’s Descent Algebra

A descent class, in the symmetric group S,, is the collection of permutations with a given descent set. It was shown by L. Solomon (J. Algebra 41 (1976), 255-264) that the product (in the group algebra Q(S,)) of two descent classes is a linear combination of descent classes. Thus descent classes generate a subalgebra of Q(.S,). We refer to it here as Solomon’s descenr algebra and denote it by C...