Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions

Abstract

In his work on P-partitions, Stembridge defined the algebra of peak functions Pi, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show that Pi is closed under coproduct, and therefore a Hopf algebra, and describe the kernel of the retraction. Billey and Haiman, in their work on Schubert polynomials, also defined a new class of quasi-symmetric functions --- shifted quasi-symmetric functions --- and we show that Pi is strictly contained in the linear span Xi of shifted quasi-symmetric functions. We show that Xi is a coalgebra, and compute the rank of the n-th graded component.

9 pages, 4 eps figures, uses epsf.sty. to be presented at FPSAC99 in Barcelona by second author

Cited by in corpus (8)

• The peak algebra and the descent algebras of types B and D
• Non-commutative Pieri operators on posets
• A lift of Schur's Q-functions to the peak algebra
• Peaks Sets of Classical Coxeter Groups
• The Hopf algebras of type B quasisymmetric functions and peak functions
• Shifted combinatorial Hopf algebras from $K$-theory
• The enriched $q$-monomial basis of the quasisymmetric functions
• Enriched $P$-partitions and peak algebras (extended abstract)