# Noncommutative symmetric functions

## Idea

Noncommutative symmetric functions are a generalisation of symmetric functions. Many concepts and ideas extend from symmetric functions to noncommutative symmetric functions and the way in which they extend sheds light on their behaviour for ordinary symmetric functions.

Noncommutative symmetric functions also arise in their own right as interesting objects of study.

## Definition

###### Definition

The graded Hopf algebra of noncommutative symmetric functions, $\NSymm$, is defined in the following way.

1. As an algebra, it is $\mathbb{Z}\langle Z_1, Z_2, \dots \rangle$, the free algebra in countably many indeterminants over $\mathbb{Z}$.
2. The comultiplication is given by $\Delta Z_n = \sum_{i + j = n} Z_i \otimes Z_j$, where $Z_0 = 1$.
3. The counit is $\epsilon(Z_n) = 0$ for $n \ge 1$.
4. The antipode is $\iota(Z_n) = \sum_{wt(\alpha) = n} (-1)^{length(\alpha)} Z_\alpha$, where $\alpha = [\alpha_1, \cdots, \alpha_k]$ is a word in $\{1,2,\dots\}$ with $length(\alpha) = k$ and $wt(\alpha) = \sum \alpha_i$.
5. The degree of $Z_n$ is $n$.

## Properties

1. The object $\NSymm$ represents a functor from not-necessarily-commutative rings to groups given by sending $B$ to formal power series in $B$ with leading term $1$, $B \mapsto 1 + t B[\![t]\!]$.

2. It is dual to the Hopf algebra of quasi-symmetric functions.

3. The Hopf algebra of symmetric functions is a quotient of $\NSymm$. The quotient mapping is given by sending $Z_i$ to the $i$th symmetric function.

### Full 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

### 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