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The Tchebyshev Transforms of the First and Second Kind
Richard Ehrenborg and Margaret Readdy
Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA
{jrge, readdy}@ms.uky.edu
Annals of Combinatorics 14 (2) pp.211-244 Summer, 2010
AMS Subject Classification: 16W30, 06A11, 06A07, 05E99
Abstract:
An in-depth study of the Tchebyshev transforms of the first and second kind of a poset is taken. The Tchebyshev transform of the first kind is shown to preserve desirable combinatorial properties, including EL-shellability and nonnegativity of the cd-index. When restricted to Eulerian posets, it corresponds to the Billera, Ehrenborg, and Readdy omega map of oriented matroids. The Tchebyshev transform of the second kind U is a Hopf algebra endomorphism on the space of quasisymmetric functions which, when restricted to Eulerian posets, coincides with Stembridge's peak enumerator. The complete spectrum of U is determined, generalizing the work of Billera, Hsiao, and van Willigenburg. The type B quasisymmetric function of a poset is introduced and, like Ehrenborg's classical quasisymmetric function of a poset, it is a comodule morphism with respect to the quasisymmetric functions QSym. Finally, similarities among the omega map, Ehrenborg's r-signed Birkhoff transform, and the Tchebyshev transforms motivate a general study of chain maps which occur naturally in the setting of combinatorial Hopf algebras.
Keywords: poset transforms, Eulerian posets, cd-index, quasisymmetric functions, Hopf algebra

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