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An introduction to quasisymmetric Schur functions. Hopf algebras, quasisymmetric functions, and Young composition tableaux. (English) Zbl 1277.16027
SpringerBriefs in Mathematics. New York, NY: Springer (ISBN 978-1-4614-7299-5/pbk; 978-1-4614-7300-8/ebook). xiv, 89 p. (2013).
To begin with, this short book is entitled “Introduction” and this has to be made clearer. Indeed this is not a textbook presenting the complete combinatorics of quasisymmetric Schur functions. But it goes far enough into the subject. The word “Introduction” is more referring to the dry listing of definitions and theorems without proofs. Providing a lot of accurate references, it is an easy and quick way to get acquainted with the subject and it gives the elements for learning more.
The first three chapters give general concepts of combinatorics (partially ordered sets, Young tableaux, Schensted insertion) and the Hopf algebras, stressing the fundamental examples of symmetric and quasisymmetric functions, Schur functions, etc. These three chapters deal with basic notions and appear like a catalogue of definitions and properties. They work as an efficient summary of the question. Chapter 4 extends notions of Young tableaux and reverse tableaux indexed by partitions given in Chapter 2 to Young composition tableaux and reverse composition tableaux indexed by compositions. Chapter 5 is a refinement of Chapter 3 connected to recent research and gives a lot of results: quasisymmetric Schur functions reliant on reverse composition tableaux and Young quasisymmetric Schur functions reliant on Young composition tableaux.
Reviewer: Marc Aubry (Nice)

16T30 Connections of Hopf algebras with combinatorics
05E05 Symmetric functions and generalizations
05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
16-02 Research exposition (monographs, survey articles) pertaining to associative rings and algebras
16T05 Hopf algebras and their applications
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