Shifted quasi-symmetric functions and the Hopf algebra of peak functions

Authors:
Nantel Bergeron;Stefan Mykytiuk;Frank Sottile;Stephanie van Willigenburg
Affiliations:
Department of Mathematics and Statistics, York University, Toronto, Ont. Canada;Department of Mathematics and Statistics, York University, Toronto, Ont. Canada;Department of Mathematics, University of Wisconsin, Van Vleck Hall, 480 Lincoln Drive, Madison, WI;Department of Mathematics and Statistics, York University, Toronto, Ont. Canada
Venue:
Discrete Mathematics
Year:
2002

Citing 0
Cited 3

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Abstract

In his work on P-partitions, Stembridge defined the algebra of peak functions Π, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show that Π 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 Π is strictly contained in the linear span Ξ of shifted quasi-symmetric functions. We show that Ξ is a coalgebra, and compute the rank of the nth graded component.