Root Systems and Dynkin Diagrams - Applications

Root systems and Dynkin diagrams appear throughout mathematics, far beyond their original context in Lie theory. Their universal nature makes them indispensable tools across many domains.

Theorem

Highest Weight Theory

Finite-dimensional irreducible representations of semisimple $\mathfrak{g}$ are in bijection with dominant integral weights $\lambda \in \mathfrak{h}^*$ satisfying $\langle \lambda, \alpha_i^\vee\rangle \in \mathbb{Z}_{\geq 0}$ for all simple roots $\alpha_i$ . The representation $V_\lambda$ has a unique highest weight vector and dimension given by the Weyl dimension formula.

This provides a complete classification of representations using combinatorial data from the root system.

Example

For $\mathfrak{sl}_3$ , dominant weights are $\lambda = m_1 \omega_1 + m_2 \omega_2$ with $m_1, m_2 \geq 0$ (where $\omega_i$ are fundamental weights). The representation $(m_1, m_2)$ appears in symmetric tensor products of the standard representation, with dimensions given by hook length formulas.

Theorem

Weyl Character Formula

The character of an irreducible representation $V_\lambda$ is: $\text{ch}(V_\lambda) = \frac{\sum_{w \in W} \epsilon(w) e^{w(\lambda + \rho)}}{\sum_{w \in W} \epsilon(w) e^{w(\rho)}}$ where $\rho = \frac{1}{2}\sum_{\alpha \in \Phi^+} \alpha$ and $\epsilon(w)$ is the sign of $w \in W$ .

This formula, expressed entirely in terms of the root system, computes multiplicities of weights in representations.

Remark

Coxeter groups and reflection groups: The Weyl groups of root systems are exactly the finite Coxeter groups. The classification of root systems thus provides a classification of finite reflection groups, with applications to geometry, combinatorics, and group theory.

Algebraic geometry applications:

Flag varieties $G/P$ are classified by subsets of simple roots
Schubert cells are indexed by Weyl group elements
Intersection cohomology uses Kazhdan-Lusztig polynomials defined from Weyl groups

Example

Singularity theory: The ADE classification (types $A_n, D_n, E_6, E_7, E_8$ ) appears in:

Simple singularities of complex curves (Du Val singularities)
Platonic solids and finite subgroups of $SU(2)$
Quiver representations and McKay correspondence
Catastrophe theory classifications

Theorem

Kostant Multiplicity Formula

The multiplicity of weight $\mu$ in the representation $V_\lambda$ is: $m_\lambda(\mu) = \sum_{w \in W} \epsilon(w) P(w(\lambda + \rho) - (\mu + \rho))$ where $P$ is the Kostant partition function counting ways to write a weight as a sum of positive roots.

Physics applications:

Grand unified theories use $E_6, E_7, E_8$ for gauge groups
String theory compactifications involve exceptional groups
Conformal field theory uses affine root systems
Integrable systems (Toda systems) are indexed by root systems

Remark

The unreasonable effectiveness of root systems stems from their capturing fundamental symmetry patterns. They encode how symmetry can be combinatorially organized, appearing whenever these patterns emerge in nature or mathematics.