# Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions

@article{Hazewinkel2002SymmetricFN, title={Symmetric Functions, Noncommutative Symmetric Functions, and Quasisymmetric Functions}, author={Michiel Hazewinkel}, journal={Acta Applicandae Mathematica}, year={2002}, volume={75}, pages={55-83} }

This paper is concerned with two generalizations of the Hopf algebra of symmetric functions that have more or less recently appeared. The Hopf algebra of noncommutative symmetric functions and its dual, the Hopf algebra of quasisymmetric functions. The focus is on the incredibly rich structure of the Hopf algebra of symmetric functions and the question of which structures and properties have good analogues for the noncommutative symmetric functions and/or the quasisymmetric functions. This…

## 32 Citations

### Symmetric Functions, Noncommutative Symmetric Functions and Quasisymmetric Functions II

- Mathematics
- 2002

Abstract
Like its precursor this paper is concerned with the Hopf algebra of noncommutative symmetric functions and its graded dual, the Hopf algebra of quasisymmetric functions. It complements and…

### An Introduction to Quasisymmetric Schur Functions: Hopf Algebras, Quasisymmetric Functions, and Young Composition Tableaux

- Mathematics
- 2013

An Introduction to Quasisymmetric Schur Functions is aimed at researchers and graduate students in algebraic combinatorics. The goal of this monograph is twofold. The first goal is to provide a…

### Word Hopf algebras

- Mathematics
- 2004

Two important generalizations of the Hopf algebra of symmetric functions are the Hopf algebra of noncommutative symmetric functions and its graded dual the Hopf algebra of quasisymmetric functions. A…

### Hopf algebras of endomorphisms of Hopf algebras

- Mathematics
- 2004

In the last decades two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves to be important: the Hopf algebra of noncommutative symmetric functions NSymm and…

### The Heisenberg product: from Hopf algebras and species to symmetric functions

- Mathematics
- 2015

Many related products and coproducts (e.g. Hadamard, Cauchy, Kronecker, induction, internal, external, Solomon, composition, Malvenuto–Reutenauer, convolution, etc.) have been defined in the…

### Nonassociative Solomon's descent algebras

- Mathematics
- 2018

Descent algebras of graded bialgebras were introduced by F. Patras as a generalization of Solomon's descent algebras for Coxeter groups of type $A$, i.e. symmetric groups. The universal enveloping…

### Renormalization groupoids in algebraic topology

- Mathematics
- 2020

Continuing work begin in arXiv:1910.12609, we interpret the Hurewicz homomorphism for Baker and Richter's noncommutative complex cobordism spectrum $M\xi$ in terms of characteristic numbers (indexed…

### A Note on Conjugation Invariants in the Dual Leibniz-Hopf Algebra

- Mathematics
- 2013

The Leibniz-Hopf algebra is the free associative algebra over Z on generators, S 1 ,S 2 ,... with coproduct given by Δ(S n )= S i ⊗ S n−i . We give odd prime and integral cases of some relations in…

### ASSOCIATIVE, LIE, AND LEFT-SYMMETRIC ALGEBRAS OF DERIVATIONS

- Mathematics
- 2014

Let Pn= k[x1, x2,…,xn] be the polynomial algebra over a field k of characteristic zero in the variables x1, x2,…,xn and ℒn be the left-symmetric Witt algebra of all derivations of Pn [Bu]. Using the…

## References

SHOWING 1-10 OF 50 REFERENCES

### The Algebra of Quasi-Symmetric Functions Is Free over the Integers

- Mathematics
- 2001

Let Z denote the Leibniz–Hopf algebra, which also turns up as the Solomon descent algebra and the algebra of noncommutative symmetric functions. As an algebra Z=Z〈Z1, Z2,…〉, the free associative…

### Duality between Quasi-Symmetrical Functions and the Solomon Descent Algebra

- Mathematics
- 1995

Abstract The ring QSym of quasi-symmetric functions is naturally the dual of the Solomon descent algebra. The product and the two coproducts of the first (extending those of the symmetric functions)…

### Hopf algebras of endomorphisms of Hopf algebras

- Mathematics
- 2004

In the last decades two generalizations of the Hopf algebra of symmetric functions have appeared and shown themselves to be important: the Hopf algebra of noncommutative symmetric functions NSymm and…

### Noncommutative Symmetric Functions Iv: Quantum Linear Groups and Hecke Algebras at q = 0

- Mathematics
- 1997

We present representation theoretical interpretations ofquasi-symmetric functions and noncommutative symmetric functions in terms ofquantum linear groups and Hecke algebras at q = 0. We obtain inthis…

### Noncommutative Symmetric Functions II: Transformations of Alphabets

- MathematicsInt. J. Algebra Comput.
- 1997

Noncommutative analogues of classical operations on symmetric functions are investigated, and several q-analogues of the Eulerian idempotents and of the Garsia-Reutenauer idempotsents are obtained.

### Noncommutative Cyclic Characters of Symmetric Groups

- MathematicsJ. Comb. Theory, Ser. A
- 1996

A multiplication formula whose commutative projection gives a combinatorial formula for the resolution of the Kronecker product of two cyclic representations of the symmetric group.

### Noncommutative Symmetric Functions V: a degenerate Version of UQ(Gln)

- MathematicsInt. J. Algebra Comput.
- 1999

We interpret quasi-symmetric functions and noncommutative symmetric functions as characters of a degenerate quantum group obtained by putting q=0 in a variant of Uq(glN).

### A Hopf-Algebra Approach to Inner Plethysm

- Mathematics
- 1994

Abstract We use the Hopf algebra structure of the algebra of symmetric functions to study the Adams operators of the complex representation rings of symmetric groups, and we give new proofs of all of…

### Symmetric functions and Hall polynomials

- Mathematics
- 1979

I. Symmetric functions II. Hall polynomials III. HallLittlewood symmetric functions IV. The characters of GLn over a finite field V. The Hecke ring of GLn over a finite field VI. Symmetric functions…

### Representations of Finite Classical Groups

- Mathematics
- 1981

The classical groups, for example the general linear or orthogonal groups, defined over finite fields, constitute a considerable part of the finite simple groups. In order to improve on the…