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

Nantel Bergeron·Stefan Mykytiuk·Frank Sottile·Stephanie van Willigenburg
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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.

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Nantel Bergeron, Stefan Mykytiuk, Frank Sottile, Stephanie van Willigenburg. 1999-04-20. Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions. https://doi.org/10.1016/s0012-365x(01)00251-5

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