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An overview of Delta -type operations on quasi-symmetric functions

Communications in Algebra (Impact Factor: 0.36). 05/2001; DOI: 10.1081/AGB-100106001
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ABSTRACT We present an overview of-type operations on the algebra of quasi-symmetric functions. Nous pr'esentons un survol de l'ensemble des propri'et'es de type-anneau de l'alg`ebre des fonctions quasi-sym'etriques. 1

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    ABSTRACT: There has been much activity in recent times in the construction and study of certain graded Hopf algebras, referred to here as combinatorial Hopf algebras. Some involve certain trees, one of which, involving rooted trees, is connected to renormalization in quantum field theory. Some involve generalizations of the Hopf algebra of symmetric functions. The book under review studies these combinatorial Hopf algebras from the unifying point of view of Coxeter complexes. These are simplicial complexes which come from a finite reflection group by passing from the real inner product space on which it acts by quotienting with the intersection of all the hyperplanes. We will describe the structure of the eight chapters of the book. Chapter 1 is an introduction to standard material on Coxeter groups, Bruhat orders and descent algebras. Important concepts are outlined. No proofs are given here (nor in Chapter 3), but references are provided. Chapter 2 is a self-contained account of left regular bands. These are semigroups that have been of recent interest in random walk theory. Many of the results on descent theory for Coxeter groups generalize to this context. Chapter 3 is on Hopf algebras, and introduces the symmetric functions, shuffles, quasi-symmetric functions and non-commutative symmetric functions. Chapter 4 gives a brief overview of the remaining Chapters 5-8, much of which represents original work of the authors. Chapter 5 on descent theory for Coxeter groups focuses on three order preserving maps defined on pairs of chambers. Chapter 6 builds on the combinatorial Hopf algebras of functions mentioned before, as well as the Hopf algebra of permutations, to construct some new combinatorial Hopf algebras. The combinatorial structures arising in this discussion include set partitions, set compositions and pairs of permutations. Chapter 7 studies the Hopf algebra indexed by pairs of permutations, showing, in particular, that it is both free and cofree. Chapter 8 relates the Hopf algebras of fully nested set compositions and of quasi-symmetric functions. Despite the formidable notational complexity, the book is well-organized and quite readable. In particular, there is a useful notation index.
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