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\begin{document}

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\section*{noncommutative symmetric function}

\hypertarget{noncommutative_symmetric_functions}{}\section*{{Noncommutative symmetric functions}}\label{noncommutative_symmetric_functions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{full_research_articles}{Full research articles}\dotfill \pageref*{full_research_articles} \linebreak
\noindent\hyperlink{long_surveys_and_lecture_notes}{Long surveys and lecture notes}\dotfill \pageref*{long_surveys_and_lecture_notes} \linebreak
\noindent\hyperlink{expositionsshort_summaries}{Expositions/short summaries}\dotfill \pageref*{expositionsshort_summaries} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{nsymm}\hypertarget{nsymm}{}
The graded [[Hopf algebra]] of \textbf{noncommutative symmetric functions}, $\NSymm$, is defined in the following way.

\begin{enumerate}%
\item As an algebra, it is $\mathbb{Z}\langle Z_1, Z_2, \dots \rangle$, the free algebra in countably many indeterminants over $\mathbb{Z}$.
\item The comultiplication is given by $\Delta Z_n = \sum_{i + j = n} Z_i \otimes Z_j$, where $Z_0 = 1$.
\item The counit is $\epsilon(Z_n) = 0$ for $n \ge 1$.
\item 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 $wt(\alpha) = \sum \alpha_i$.
\item The degree of $Z_n$ is $n$.

\end{enumerate}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{enumerate}%
\item 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 $1$, $B \mapsto 1 + t B[\![t]\!]$.


\item It is dual to the Hopf algebra of [[quasi-symmetric functions]].


\item 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.



\end{enumerate}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{full_research_articles}{}\subsubsection*{{Full research articles}}\label{full_research_articles}

\begin{itemize}%
\item G. Duchamp, F. Hivert, J.-Y. Thibon, \emph{Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras}, Internat. J. Alg. Comput. 12 (2002), 671--717.
\item [[I. M. Gelfand]], D. Krob, A. Lascoux, B. Leclerc, V. S. Retakh, J.-Y. Thibon, \emph{Noncommutative symmetric functions}, Adv. in Math. \textbf{112} (1995), 218--348, \href{http://arxiv.org/abs/hep-th/9407124}{hep-th/9407124}
\item Jean-Christophe Novelli, Jean-Yves Thibon, \emph{Noncommutative symmetric functions and Lagrange inversion}, \href{http://arxiv.org/abs/math/0512570}{math.CO/0512570}; \emph{Noncommutative symmetric functions and an amazing matrix} \href{http://arxiv.org/abs/1109.1184}{arxiv/1109.1184}
\item Lenny Tevlin, \emph{Noncommutative Monomial Symmetric Functions}, Formal Power Series and Algebraic Combinatorics Nankai University, Tianjin, China, 2007, proceedings \href{http://www.fpsac.org/FPSAC07/SITE07/PDF-Proceedings/Talks/83.pdf}{pdf}
\item D. Krob, J.-Y. Thibon, \emph{Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at $q = 0$}, \href{http://hal.inria.fr/docs/00/05/79/10/PDF/ncsf4.pdf}{pdf}

\end{itemize}
\hypertarget{long_surveys_and_lecture_notes}{}\subsubsection*{{Long surveys and lecture notes}}\label{long_surveys_and_lecture_notes}

\begin{itemize}%
\item Michael Hazewinkel, \emph{Symmetric functions, noncommutative symmetric functions and quasisymmetric functions}, \href{http://oai.cwi.nl/oai/asset/10344/10344B.pdf}{pdf}
\item [[V. Retakh]] and R. Wilson, Advanced Course on Quasideterminants and Universal Localization: \href{http://castellet.cat/Publications/quaderns/Quadern41.pdf}{pdf} (see the part \emph{Factorization of Noncommutative Polynomials and Noncommutative Symmetric Functions})

\end{itemize}
\hypertarget{expositionsshort_summaries}{}\subsubsection*{{Expositions/short summaries}}\label{expositionsshort_summaries}

\begin{itemize}%
\item Mike Zabrocki, \emph{Non-commutative symmetric functions II: Combinatorics and coinvariants}, slides from a talk \href{http://garsia.math.yorku.ca/~zabrocki/talks/coinvariants.pdf}{pdf}, III: A representation theoretical approach \href{http://garsia.math.yorku.ca/~zabrocki/talks/reptheory.pdf}{pdf}
\item Lenny Tevlin, \emph{Introduction to quasisymmetric and noncommutative symmetric functions}, slides, Fields Institute 2010 \href{http://www.fields.utoronto.ca/programs/scientific/10-11/schubert/Noncommutative-Symmetric-and-Quasi-Symmetric-Functions-Fields-2010.pdf}{pdf}

\end{itemize}
[[!redirects noncommutative symmetric function]] [[!redirects noncommutative symmetric functions]]



\end{document}