The Tchebyshev transforms of the first and second kind

Richard Ehrenborg; Margaret Readdy

doi:10.1007/s00026-010-0057-2

The Tchebyshev transforms of the first and second kind

Richard Ehrenborg, Margaret Readdy

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

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.

Original language	English
Pages (from-to)	211-244
Number of pages	34
Journal	Annals of Combinatorics
Volume	14
Issue number	2
DOIs	https://doi.org/10.1007/s00026-010-0057-2
State	Published - Jun 2010

Bibliographical note

Funding Information:
Acknowledgments. We graciously thank Gábor Hetyei for inspiring us to study the Tcheby-shev transform and suggesting research directions. We thank Ira Gessel for directing us to Chak-On Chow’s work on the type B quasisymmetric functions. The authors also thank the referees for their comments and suggestions. The first author was partially supported by National Science Foundation grant 0200624 and by a University of Kentucky College of Arts & Sciences Faculty Research Fellowship. The second author was partially supported by a University of Kentucky College of Arts & Sciences Research Grant. Both authors thank the Institute for Advanced Study/Park City Mathematics Institute for providing a stimulating work environment where this work was initiated and MIT, where the authors finished this work while on sabbatical.

Funding

Acknowledgments. We graciously thank Gábor Hetyei for inspiring us to study the Tcheby-shev transform and suggesting research directions. We thank Ira Gessel for directing us to Chak-On Chow’s work on the type B quasisymmetric functions. The authors also thank the referees for their comments and suggestions. The first author was partially supported by National Science Foundation grant 0200624 and by a University of Kentucky College of Arts & Sciences Faculty Research Fellowship. The second author was partially supported by a University of Kentucky College of Arts & Sciences Research Grant. Both authors thank the Institute for Advanced Study/Park City Mathematics Institute for providing a stimulating work environment where this work was initiated and MIT, where the authors finished this work while on sabbatical.

Funders	Funder number
University of Kentucky College of Arts & Sciences
University of Kentucky College of Arts & Sciences Research
National Science Foundation Arctic Social Science Program
Directorate for Mathematical and Physical Sciences	0200624

Keywords

Eulerian posets
Hopf algebra
Poset transforms
Quasisymmetric functions
cd-index

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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10.1007/s00026-010-0057-2

Cite this

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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.",

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The Tchebyshev transforms of the first and second kind

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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