Combinatorial Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 17 September 2013

Appendix A

A8. Roots, weights and paths

To each of the “types”, $A_n,$ $B_n,$ etc., there is an associated hyperplane arrangement $𝒜$ in ${ℝ}^n\text{.}$ $$\begin{array}{c} \\ \text{Hyperplane arrangement for}\hspace{0.17em}{A}_2\end{array}$$ The space ${ℝ}^n$ has the usual Euclidean inner product $⟨,⟩\text{.}$ For each hyperplane in the arrangement $𝒜$ 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 $𝒜\text{.}$ $$\begin{array}{c} \\ \text{Root system for}\hspace{0.17em}{A}_2\end{array}$$ 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 ${ℝ}^n\backslash {\bigcup }_{H\in 𝒜}H\text{.}$ $$\begin{array}{c} \\ \text{A chamber for}\hspace{0.17em}{A}_2\end{array}$$ 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 $$\alpha >0,\text{if}\hspace{0.17em}\alpha \hspace{0.17em}\text{is a positive root,}\text{and}\alpha <0,\text{if}\hspace{0.17em}\alpha \hspace{0.17em}\text{is a negative root.}$$ 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 ${ℝ}^n$ such that $$⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij},\text{where}{\alpha }_j^{\vee }=\frac{2{\alpha }_j}{⟨{\alpha }_j,{\alpha }_j⟩}\text{.}$$ Then $$P=\sum _{i=1}^r\mathbb{Z}{\omega }_i,\text{and}{P}^{+}=\sum _{i=1}^n\mathbb{N}{\omega }_i,\text{where}\mathbb{N}={\mathbb{Z}}_{\ge 0},$$ are the lattice of integral weights and the cone of dominant integral weights, respectively. $$\begin{array}{cc} & \\ \text{Lattice of integral weights}& \text{Cone of dominant integral weights}\end{array}$$ There is a one-to-one correspondence between the irreducible representations of $𝔤$ 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 $$\begin{array}{c}\begin{array}{cccc}{\pi }_{\lambda }:& [0,1]& ⟶& {ℝ}^n\\ & t& ⟼& t\lambda \text{.}\end{array}\\ λ \\ \text{Path from}\hspace{0.17em}0\hspace{0.17em}\text{to}\hspace{0.17em}\lambda \end{array}$$ The set $𝒫{\pi }_{\lambda }$ is given by $$𝒫{\pi }_{\lambda }=\left\{f_{i_1}\cdots f_{i_k}{\pi }_{\lambda }\hspace{0.17em}\right|\hspace{0.17em}1\le i_1,\dots ,i_k\le n\}$$ where $f_1,\dots ,f_n$ are the path operators introduced in [Lit1995]. These paths might look like $$\begin{array}{c} \\ \text{Path in}\hspace{0.17em}𝒫{\pi }_{\lambda }\end{array}$$ 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].

Notes and references

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..

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