# Quasi-symmetric functions

## Idea

Quasi symmetric functions are a generalisation of symmetric functions and are closely related to noncommutative symmetric functions.

## Definition

###### Definition

Let $X$ be a totally ordered set of indeterminants. Let $R$ be a ring. A polynomial in $R[X]$ or a power series in $R[ [X] ]$ is said to be quasi-symmetric if whenever $X_1 \lt X_2 \lt \dots \lt X_n$ and $Y_1 \lt Y_2 \lt \dots \lt Y_n$ are finite sets of indeterminants then the coefficients of $X_1^{i_1} X_2^{i_2} \cdots X_n^{i_n}$ and $Y_1^{i_1} Y_2^{i_2} \cdots Y_n^{i_n}$ are the same.

###### Definition

The ring $\QSymm^{\hat{}}$ is defined as the ring of quasi-symmetric power series over $\mathbb{Z}$ in countably many variables. Its subring $\QSymm$ is defined as the ring of quasi-symmetric polynomials (meaning, power series of bounded degree).

(Copied from noncommutative symmetric function as the two concepts are often studied together.)

### Research articles

• G. Duchamp, F. Hivert, J.-Y. Thibon, Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. Alg. Comput. 12 (2002), 671–717.
• I. M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, J.-Y. Thibon, Noncommutative symmetric functions, Adv. in Math. 112 (1995), 218–348, hep-th/9407124
• Jean-Christophe Novelli, Jean-Yves Thibon, Noncommutative symmetric functions and Lagrange inversion, math.CO/0512570; Noncommutative symmetric functions and an amazing matrix arxiv/1109.1184
• Lenny Tevlin, Noncommutative Monomial Symmetric Functions, Formal Power Series and Algebraic Combinatorics Nankai University, Tianjin, China, 2007, proceedings pdf
• D. Krob, J.-Y. Thibon, Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at $q = 0$, pdf
• Christos A. Athanasiadis, Power sum expansion of chromatic quasisymmetric functions, arxiv/1409.2595

### Long surveys and lecture notes

• Michael Hazewinkel, Symmetric functions, noncommutative symmetric functions and quasisymmetric functions, pdf
• V. Retakh and R. Wilson, Advanced Course on Quasideterminants and Universal Localization: pdf (see the part Factorization of Noncommutative Polynomials

and Noncommutative Symmetric Functions_)

### Expositions/short summaries

• Mike Zabrocki, Non-commutative symmetric functions II: Combinatorics and coinvariants, slides from a talk pdf, III: A representation theoretical approach pdf
• Lenny Tevlin, Introduction to quasisymmetric and noncommutative symmetric functions, slides, Fields Institute 2010 pdf
category: combinatorics