\documentclass[12pt,titlepage]{article}

\usepackage{amsmath}
\usepackage{mathrsfs}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage{graphicx}
\usepackage{color}
\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{xparse}
\usepackage{hyperref}

%----Macros----------
%
% Unresolved issues:
%
%  \righttoleftarrow
%  \lefttorightarrow
%
%  \color{} with HTML colorspec
%  \bgcolor
%  \array with options (without options, it's equivalent to the matrix environment)

% Of the standard HTML named colors, white, black, red, green, blue and yellow
% are predefined in the color package. Here are the rest.
\definecolor{aqua}{rgb}{0, 1.0, 1.0}
\definecolor{fuschia}{rgb}{1.0, 0, 1.0}
\definecolor{gray}{rgb}{0.502, 0.502, 0.502}
\definecolor{lime}{rgb}{0, 1.0, 0}
\definecolor{maroon}{rgb}{0.502, 0, 0}
\definecolor{navy}{rgb}{0, 0, 0.502}
\definecolor{olive}{rgb}{0.502, 0.502, 0}
\definecolor{purple}{rgb}{0.502, 0, 0.502}
\definecolor{silver}{rgb}{0.753, 0.753, 0.753}
\definecolor{teal}{rgb}{0, 0.502, 0.502}

% Because of conflicts, \space and \mathop are converted to
% \itexspace and \operatorname during preprocessing.

% itex: \space{ht}{dp}{wd}
%
% Height and baseline depth measurements are in units of tenths of an ex while
% the width is measured in tenths of an em.
\makeatletter
\newdimen\itex@wd%
\newdimen\itex@dp%
\newdimen\itex@thd%
\def\itexspace#1#2#3{\itex@wd=#3em%
\itex@wd=0.1\itex@wd%
\itex@dp=#2ex%
\itex@dp=0.1\itex@dp%
\itex@thd=#1ex%
\itex@thd=0.1\itex@thd%
\advance\itex@thd\the\itex@dp%
\makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}}
\makeatother

% \tensor and \multiscript
\makeatletter
\newif\if@sup
\newtoks\@sups
\def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}%
\def\reset@sup{\@supfalse\@sups={}}%
\def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else%
  \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}%
  \else \append@sup#2 \@suptrue \fi%
  \expandafter\mk@scripts\fi}
\def\tensor#1#2{\reset@sup#1\mk@scripts#2_/}
\def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2%
  \reset@sup\mk@scripts#3_/}
\makeatother

% \slash
\makeatletter
\newbox\slashbox \setbox\slashbox=\hbox{$/$}
\def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$}
  \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa
  \copy\slashbox \kern-\@tempdima \box\@tempboxa}
\def\slash{\protect\itex@pslash}
\makeatother

% math-mode versions of \rlap, etc
% from Alexander Perlis, "A complement to \smash, \llap, and lap"
%   http://math.arizona.edu/~aprl/publications/mathclap/
\def\clap#1{\hbox to 0pt{\hss#1\hss}}
\def\mathllap{\mathpalette\mathllapinternal}
\def\mathrlap{\mathpalette\mathrlapinternal}
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}
\def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}}

% Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2}
\let\oldroot\root
\def\root#1#2{\oldroot #1 \of{#2}}
\renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}}

% Manually declare the txfonts symbolsC font
\DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n}
\SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n}
\DeclareFontSubstitution{U}{txsyc}{m}{n}

% Manually declare the stmaryrd font
\DeclareSymbolFont{stmry}{U}{stmry}{m}{n}
\SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n}

% Manually declare the MnSymbolE font
\DeclareFontFamily{OMX}{MnSymbolE}{}
\DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n}
\SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n}
\DeclareFontShape{OMX}{MnSymbolE}{m}{n}{
    <-6>  MnSymbolE5
   <6-7>  MnSymbolE6
   <7-8>  MnSymbolE7
   <8-9>  MnSymbolE8
   <9-10> MnSymbolE9
  <10-12> MnSymbolE10
  <12->   MnSymbolE12}{}

% Declare specific arrows from txfonts without loading the full package
\makeatletter
\def\re@DeclareMathSymbol#1#2#3#4{%
    \let#1=\undefined
    \DeclareMathSymbol{#1}{#2}{#3}{#4}}
\re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116}
\re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117}
\re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118}
\re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119}
\re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46}
\re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121}
\re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12}
\re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64}
\re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6}
\re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77}
\re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77}
\makeatother

% \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE
\makeatletter
\def\Decl@Mn@Delim#1#2#3#4{%
  \if\relax\noexpand#1%
    \let#1\undefined
  \fi
  \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}}
\def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}}
\def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}}
\Decl@Mn@Open{\llangle}{mnomx}{'164}
\Decl@Mn@Close{\rrangle}{mnomx}{'171}
\Decl@Mn@Open{\lmoustache}{mnomx}{'245}
\Decl@Mn@Close{\rmoustache}{mnomx}{'244}
\makeatother

% Widecheck
\makeatletter
\DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}}
\def\@widecheck#1#2{%
    \setbox\z@\hbox{\m@th$#1#2$}%
    \setbox\tw@\hbox{\m@th$#1%
       \widehat{%
          \vrule\@width\z@\@height\ht\z@
          \vrule\@height\z@\@width\wd\z@}$}%
    \dp\tw@-\ht\z@
    \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@
    \setbox\tw@\hbox{%
       \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box
\tw@}}}%
    {\ooalign{\box\tw@ \cr \box\z@}}}
\makeatother

% \mathraisebox{voffset}[height][depth]{something}
\makeatletter
\NewDocumentCommand\mathraisebox{moom}{%
\IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{%
\IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}%
}{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}%
\mathpalette\@temp{#4}}
\makeatletter

% udots (taken from yhmath)
\makeatletter
\def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.}
\mkern2mu\raise4\p@\hbox{.}\mkern1mu
\raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}}
\makeatother

%% Fix array
\newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}}
%% \itexnum is a noop
\newcommand{\itexnum}[1]{#1}

%% Renaming existing commands
\newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}}
\newcommand{\widevec}{\overrightarrow}
\newcommand{\darr}{\downarrow}
\newcommand{\nearr}{\nearrow}
\newcommand{\nwarr}{\nwarrow}
\newcommand{\searr}{\searrow}
\newcommand{\swarr}{\swarrow}
\newcommand{\curvearrowbotright}{\curvearrowright}
\newcommand{\uparr}{\uparrow}
\newcommand{\downuparrow}{\updownarrow}
\newcommand{\duparr}{\updownarrow}
\newcommand{\updarr}{\updownarrow}
\newcommand{\gt}{>}
\newcommand{\lt}{<}
\newcommand{\map}{\mapsto}
\newcommand{\embedsin}{\hookrightarrow}
\newcommand{\Alpha}{A}
\newcommand{\Beta}{B}
\newcommand{\Zeta}{Z}
\newcommand{\Eta}{H}
\newcommand{\Iota}{I}
\newcommand{\Kappa}{K}
\newcommand{\Mu}{M}
\newcommand{\Nu}{N}
\newcommand{\Rho}{P}
\newcommand{\Tau}{T}
\newcommand{\Upsi}{\Upsilon}
\newcommand{\omicron}{o}
\newcommand{\lang}{\langle}
\newcommand{\rang}{\rangle}
\newcommand{\Union}{\bigcup}
\newcommand{\Intersection}{\bigcap}
\newcommand{\Oplus}{\bigoplus}
\newcommand{\Otimes}{\bigotimes}
\newcommand{\Wedge}{\bigwedge}
\newcommand{\Vee}{\bigvee}
\newcommand{\coproduct}{\coprod}
\newcommand{\product}{\prod}
\newcommand{\closure}{\overline}
\newcommand{\integral}{\int}
\newcommand{\doubleintegral}{\iint}
\newcommand{\tripleintegral}{\iiint}
\newcommand{\quadrupleintegral}{\iiiint}
\newcommand{\conint}{\oint}
\newcommand{\contourintegral}{\oint}
\newcommand{\infinity}{\infty}
\newcommand{\bottom}{\bot}
\newcommand{\minusb}{\boxminus}
\newcommand{\plusb}{\boxplus}
\newcommand{\timesb}{\boxtimes}
\newcommand{\intersection}{\cap}
\newcommand{\union}{\cup}
\newcommand{\Del}{\nabla}
\newcommand{\odash}{\circleddash}
\newcommand{\negspace}{\!}
\newcommand{\widebar}{\overline}
\newcommand{\textsize}{\normalsize}
\renewcommand{\scriptsize}{\scriptstyle}
\newcommand{\scriptscriptsize}{\scriptscriptstyle}
\newcommand{\mathfr}{\mathfrak}
\newcommand{\statusline}[2]{#2}
\newcommand{\tooltip}[2]{#2}
\newcommand{\toggle}[2]{#2}

% Theorem Environments
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\newtheorem*{utheorem}{Theorem}
\newtheorem*{ulemma}{Lemma}
\newtheorem*{uprop}{Proposition}
\newtheorem*{ucor}{Corollary}
\theoremstyle{definition}
\newtheorem{defn}{Definition}
\newtheorem{example}{Example}
\newtheorem*{udefn}{Definition}
\newtheorem*{uexample}{Example}
\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem{note}{Note}
\newtheorem*{uremark}{Remark}
\newtheorem*{unote}{Note}

%-------------------------------------------------------------------

\begin{document}

%-------------------------------------------------------------------


\section*{symmetric function}

\hypertarget{symmetric_functions}{}\section*{{Symmetric functions}}\label{symmetric_functions}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

A \emph{symmetric function} is roughly a [[polynomial]] that is invariant under [[permutation]] of its [[variables]]. However, this is only strictly correct if the number of variables is [[finite number|finite]], while symmetric functions depend on a countably infinite number of variables. The only symmetric \emph{polynomials} in infinitely many variables are the constants. To fix this, one allows infinitely many terms, as long as the \emph{degree} is finite.

For example, there is a $1$-dimensional space of homogeneous symmetric functions of degree $1$, with basis

\begin{displaymath}
x_1 + x_2 + \cdots = \sum_i x_i .
\end{displaymath}
There is a $2$-dimensional space of homogeneous symmetric functions of degree $2$, with basis

\begin{displaymath}
x_1^2 + x_2^2 + \cdots = \sum_i x_i^2
\end{displaymath}
and

\begin{displaymath}
\sum_{i \lt j} x_i x_j .
\end{displaymath}
These basis elementa are called the \emph{[[elementary symmetric functions]]}.

(The homogeneous symmetric functions of degree $0$ are just the constants, as usual.)

There is also a noncommutative analogue: [[noncommutative symmetric functions]].

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Let $\Lambda_n$ be the [[ring]] consisting of [[polynomials]] in $n$ [[variables]] $x_1, \dots, x_n$ that are invariant under all [[permutations]] of the variables; these are the \textbf{symmetric functions in $n$ variables} or \textbf{symmetric polynomials} in $n$ variables. The rings $\Lambda_n$ are [[graded ring|graded]] by degree in the usual way, and there are homomorphisms of graded rings

\begin{displaymath}
\Lambda_{n+1} \hookrightarrow \Lambda_{n} ,
\end{displaymath}
given by setting the $(n+1)$st variable equal to zero. Taking the [[limit]] (in the category-theoretic sense) of these rings $\Lambda_n$ as $n \to \infty$, we get the graded ring $\Lambda_\infty$. This is usually called $\Lambda$, or the ring of \textbf{symmetric functions in countably many variables}, or simply \textbf{symmetric functions} (or even \textbf{symmetric polynomials}, even though very few of them are really polynomials).

Alternatively, $\Lambda$ can be constructed as a [[colimit]] of rings using the homomorphisms

\begin{displaymath}
\Lambda_{n} \hookrightarrow \Lambda_{n+1} ,
\end{displaymath}
where we add in new terms with the new variable to make the result symmetric.

The definition depends on the [[ground field|ground]] [[field]] (or [[commutative ring]] or [[rig]]) $k$, so we may write $\Lambda(k)$ to be precise.

A query about the Hazewinkel's description of the construction of $\Lambda$ as a (co)limit is \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=3117&Focus=25812#Comment_25812}{here}.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

Symmetric functions play a fundamental role throughout [[representation theory]], [[combinatorics]] and [[algebraic topology]]. The ring of symmetric functions, $\Lambda$, has many interesting properties. For example, it is the free $\lambda$-[[lambda-ring|ring]] on one generator (a coincidence of notation that has not been ignored). It is also a [[plethory]]. There are various bases of $\Lambda$ whose elements are in natural one-to-one correspondence with [[Young diagram|Young diagrams]].

$\Lambda$ acquires many of these properties from the fact that it is the [[Grothendieck ring]] of the category of $k$-linear [[species]]. This category is defined to be the functor category

\begin{displaymath}
[\mathbb{P}, Vect_{k}]
\end{displaymath}
where $\mathbb{P}$ is the groupoid of [[finite sets]] and bijections, and $Vect_k$ is the category of vector spaces over a field $k$ of characteristic zero. The category of $k$-linear species becomes a symmetric monoidal category thanks to [[Day convolution]]. It is also a [[semisimple category|semisimple abelian category]], with a basis of objects given by irreducible [[representation|representations]] of [[symmetric groups]] --- one for each [[Young diagram]]. So, the [[Grothendieck group]] of this category becomes a commutative ring with a basis given by Young diagrams, and this is just $\Lambda$.

$\Lambda$ is also the Grothendieck ring of a somewhat smaller $Schur$, whose objects are [[Schur functors]]. There are various ways to think about these, but they can be identified with $k$-linear species

\begin{displaymath}
F: \mathbb{P} \to Vect_{k}
\end{displaymath}
with the special property that $F(n)$ is finite-dimensional for all $n$ and $0$ for $n$ sufficiently large. The category of Schur functors is again a [[semisimple category|semisimple abelian category]] with a basis of objects given by irreducible [[representation|representations]] of [[symmetric groups]], so its Grothendieck ring is again $\Lambda$. For more on this, see [[Schur functor]].

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item In the context of [[universal characteristic classes]] of [[vector bundles]], the [[splitting principle]] identifies higher [[Chern classes]] with symmetric polynomials in copies of the [[first Chern class]].

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Witt vector]]

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

\begin{itemize}%
\item [[Michiel Hazewinkel]], \emph{[[Hazewinkel, Witt vectors|Witt vectors]]}, Part 1. \href{http://arxiv.org/abs/0804.3888}{arXiv/0804.3888}

\end{itemize}
The classification described above of irreducible $S_n$-modules over $\mathbb{C}$ also works unchanged for any algebraically closed field $k$ of characteristic zero. It also works when $k$ has characteristic $p$ and $n$ is not divisible by $p$. There's an exercise at the end of section 6.1 of Serre's book (page 64 of the french edition) that says that if $k$ is a field of characteristic $p \gt 0$ then the group algebra of the group $G$ is semisimple iff $p$ doesn't divide the order of $G$.

Apart from this, the field matters a lot. There is a construction that gives all irreducible $k S_n$-modules for any field $k$, field, completely analogous to the Specht module construction over $\mathbb{C}$. However, it describes each module as a quotient module of a Specht module, and in general not even the dimension of these irreducible modules is known, let alone an explicit basis, or representing matrices.

The following article surveys the subject in light of the connections to [[Hopf algebra]]s and also to [[noncommutative symmetric function|noncommutative analogue]]:

\begin{itemize}%
\item [[Michiel Hazewinkel]], \emph{Symmetric functions, noncommutative symmetric functions and quasisymmetric functions}, \href{http://oai.cwi.nl/oai/asset/10344/10344B.pdf}{pdf}

\end{itemize}
Among the best books in the subject are

\begin{itemize}%
\item Gordon D. James, \emph{The representation theory of the symmetric groups}, Lecture Notes in Mathematics \textbf{682}, Springer, Berlin, 1978.
\item Gordon James, Adalbert Kerber, \emph{The representation theory of the symmetric group}, With a foreword by P. M. Cohn. With an intr. by G. de B. Robinson. Enc. of Math. and its Appl. \textbf{16}, Addison-Wesley 1981. xxviii+510 pp.

\end{itemize}
James also has a readable survey article that outlines developments in the `80's and `90's:

\begin{itemize}%
\item Gordon D. James, Symmetric groups and Schur algebras, in \emph{Algebraic Groups and their Representations}, eds. R. W. Carter and J. Saxl, Kluwer Acad. Publ. Netherlands 1998.

\end{itemize}
Another approach is described here:

\begin{itemize}%
\item Alexander Kleshchev, \emph{Linear and Projective Representations of Symmetric Groups}, Cambridge University Press, Cambridge 2009.

\end{itemize}
One can study symmetric functions in any characteristic, or over any integral domain. The power sum symmetric functions do not generate the ring of symmetric functions over $\mathbb{Z}$, and this difference matters. They appear to be of limited usefulness in the description of the modular irreducible representations of $S_n$.

[[!redirects symmetric function]] [[!redirects symmetric functions]] [[!redirects symmetric polynomial]] [[!redirects symmetric polynomials]]



\end{document}