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Mathematics  2005 

Enriched $P$-partitions and peak algebras (extended abstract)

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Abstract:

We generalize Stembridge's enriched $P$-partitions and use this theory to outline the structure of peak algebras for the symmetric group and the hyperoctahedral group. Whereas Stembridge's enriched $P$-partitions are related to quasisymmetric functions (the coalgebra dual to Solomon's type A descent algebra), our generalized enriched $P$-partitions are related to type B quasisymmetric functions (the coalgebra dual to Solomon's type B descent algebra). Using these functions, we explore three different peak algebras: the "interior" and "left" peak algebras of type A, and a new type B peak algebra. Our results specialize to results for commutative peak algebras as well.

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