Last update: 17 September 2013

To each of the “types”, ${A}_{n},$ ${B}_{n},$
etc., there is an associated hyperplane arrangement $\mathcal{A}$ in ${\mathbb{R}}^{n}\text{.}$
$$\begin{array}{c}\n\n\n\n\n\n\\ \text{Hyperplane arrangement for}\hspace{0.17em}{A}_{2}\end{array}$$
The space ${\mathbb{R}}^{n}$ has the usual Euclidean inner product $\u27e8,\u27e9\text{.}$
For each hyperplane in the arrangement $\mathcal{A}$ we choose two vectors orthogonal to the hyperplane and pointing in opposite directions.
This set of chosen vectors is called the *root system* $R$ associated to $\mathcal{A}\text{.}$
$$\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\\ \text{Root system for}\hspace{0.17em}{A}_{2}\end{array}$$
There is a convention for choosing the lengths of these vectors but we shall not worry about that here.

Choose a chamber (connected component) $C$ of ${\mathbb{R}}^{n}\backslash {\bigcup}_{H\in \mathcal{A}}H\text{.}$
$$\begin{array}{c}\n\n\n\n\n\n\n\\ \text{A chamber for}\hspace{0.17em}{A}_{2}\end{array}$$
For each root $\alpha \in R$ we say that $\alpha $ is *positive* if it points toward the same
side of the hyperplane as $C$ is and *negative* if points toward the opposite side. It is standard notation to write
$$\alpha >0,\phantom{\rule{2em}{0ex}}\text{if}\hspace{0.17em}\alpha \hspace{0.17em}\text{is a positive root,}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\alpha <0,\phantom{\rule{2em}{0ex}}\text{if}\hspace{0.17em}\alpha \hspace{0.17em}\text{is a negative root.}$$
The positive roots which are associated to hyperplanes which form the walls of $C$ are the *simple roots*
$\{{\alpha}_{1},\dots ,{\alpha}_{n}\}\text{.}$
The *fundamental weights* are the vectors $\{{\omega}_{1},\dots ,{\omega}_{n}\}$
in ${\mathbb{R}}^{n}$ such that
$$\u27e8{\omega}_{i},{\alpha}_{j}^{\vee}\u27e9={\delta}_{ij},\phantom{\rule{1em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}{\alpha}_{j}^{\vee}=\frac{2{\alpha}_{j}}{\u27e8{\alpha}_{j},{\alpha}_{j}\u27e9}\text{.}$$
Then
$$P=\sum _{i=1}^{r}\mathbb{Z}{\omega}_{i},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{P}^{+}=\sum _{i=1}^{n}\mathbb{N}{\omega}_{i},\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{1em}{0ex}}\mathbb{N}={\mathbb{Z}}_{\ge 0},$$
are the lattice of *integral weights* and the cone of *dominant integral weights*, respectively.
$$\begin{array}{cc}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n& \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ \text{Lattice of integral weights}& \text{Cone of dominant integral weights}\end{array}$$
There is a one-to-one correspondence between the irreducible representations of $\U0001d524$ and the elements of the cone
${P}^{+}$ in the lattice $P\text{.}$

Let $\lambda $ be a point in ${P}^{+}\text{.}$ Then the straight line path from $0$ to $\lambda $ is the map $$\begin{array}{c}\begin{array}{cccc}{\pi}_{\lambda}:& [0,1]& \u27f6& {\mathbb{R}}^{n}\\ & t& \u27fc& t\lambda \text{.}\end{array}\\ \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\lambda \n\n\\ \text{Path from}\hspace{0.17em}0\hspace{0.17em}\text{to}\hspace{0.17em}\lambda \end{array}$$ The set $\mathcal{P}{\pi}_{\lambda}$ is given by $$\mathcal{P}{\pi}_{\lambda}=\left\{{f}_{{i}_{1}}\cdots {f}_{{i}_{k}}{\pi}_{\lambda}\hspace{0.17em}\right|\hspace{0.17em}1\le {i}_{1},\dots ,{i}_{k}\le n\}$$ where ${f}_{1},\dots ,{f}_{n}$ are the path operators introduced in [Lit1995]. These paths might look like $$\begin{array}{c}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ \text{Path in}\hspace{0.17em}\mathcal{P}{\pi}_{\lambda}\end{array}$$ They are always piecewise linear and end in a point in $P\text{.}$

**References**

The basics of root systems can be found in [Hum1978]. The reference for the path model of Littelmann is [Lit1995].

This is the survey paper *Combinatorial Representation Theory*, written by Hélène Barcelo and Arun Ram.

*Key words and phrases.* Algebraic combinatorics, representations.

Barcelo was supported in part by National Science Foundation grant DMS-9510655.

Ram was supported in part by National Science Foundation grant DMS-9622985.

This paper was written while both authors were in residence at MSRI. We are grateful for the hospitality and financial support of MSRI..