[Paper Review] On the Dynkin index of a principal $\mathfrak{sl}_2$-subalgebra

Dmitri I. Panyushev|ArXiv.org|Mar 2, 2009

Advanced Topics in Algebra4 references3 citations

TL;DR

This paper establishes a closed-form formula for the Dynkin index of a principal $ \mathfrak{sl}_2$-subalgebra in any simple Lie algebra ${\mathfrak{g}}$, using the Freudenthal–de Vries 'strange formula'. The key result expresses the index as $\frac{\dim{\mathfrak{g}}}{6}h^*({\mathfrak{g}}^\vee)r$, where $h^*({\mathfrak{g}}^\vee)$ is the dual Coxeter number of the Langlands dual ${\mathfrak{g}}^\vee$ and $r$ is the rank. This enables explicit computation of Dynkin indices for irreducible representations of ${\mathfrak{g}}$ when viewed as $\mathfrak{sl}_2$-modules and yields a new identity linking exponents of ${\mathfrak{g}}$ to dual Coxeter numbers of ${\mathfrak{g}}$ and ${\mathfrak{g}}^\vee$. The formula is uniform across all types and invariant under unfolding of Dynkin diagrams.

ABSTRACT

Let $g$ be a simple Lie algebra over an algebraically closed field of characteristic zero. The goal of this note is to prove a closed formula for the Dynkin index of a principal $sl_2$-subalgebra of $g$. The key step in the proof uses the "strange formula" of Freudenthal--de Vries. As an application, we (1) compute the Dynkin index any simple $g$-module regarded as $sl_2$-module and (2) obtain an identity connecting the exponents of $g$ and the dual Coxeter numbers of both $g$ and the Langlands dual $g^\vee$.

Motivation & Objective

• To derive a closed-form expression for the Dynkin index of a principal $\mathfrak{sl}_2$-subalgebra in any simple Lie algebra ${\mathfrak{g}}$.
• To apply this formula to compute the Dynkin index of any irreducible ${\mathfrak{g}}$-module when viewed as a $\mathfrak{sl}_2$-module.
• To establish a new identity connecting the exponents of ${\mathfrak{g}}$ with the dual Coxeter numbers of ${\mathfrak{g}}$ and its Langlands dual ${\mathfrak{g}}^\vee$.

Proposed method

• Utilizes the Freudenthal–de Vries 'strange formula' to relate the sum of squared heights of positive roots to the dimension and dual Coxeter number of ${\mathfrak{g}}$.
• Applies the multiplicativity property of the Dynkin index for nested subalgebras: ${\mathsf{ind}}({\mathfrak{h}}\subset{\mathfrak{g}}) = {\mathsf{ind}}({\mathfrak{h}}\subset{\mathfrak{s}}) \cdot {\mathsf{ind}}({\mathfrak{s}}\subset{\mathfrak{g}})$.
• Expresses the Dynkin index of the principal $\mathfrak{sl}_2$-subalgebra as $\frac{\dim{\mathfrak{g}}}{6}h^*({\mathfrak{g}}^\vee)r$ using the strange formula and root height sums.
• Uses the additivity of the Dynkin index for direct sums of representations to compute indices of irreducible ${\mathfrak{g}}$-modules as $\mathfrak{sl}_2$-modules.
• Relies on the standard normalization where $(\theta,\theta)_{\mathfrak{g}} = 2$, ensuring consistency in bilinear form scaling.
• Applies the formula to derive a new identity involving the sum of binomial coefficients $\binom{2m_i + 2}{3}$ over the exponents $m_i$ of ${\mathfrak{g}}$.

Experimental results

Research questions

• RQ1 What is the closed-form expression for the Dynkin index of a principal $\mathfrak{sl}_2$-subalgebra in a simple Lie algebra ${\mathfrak{g}}$?
• RQ2 How can this index be used to compute the Dynkin index of any irreducible ${\mathfrak{g}}$-module when restricted to the principal $\mathfrak{sl}_2$-subalgebra?
• RQ3 Is there a uniform identity connecting the exponents of ${\mathfrak{g}}$ and the dual Coxeter numbers of ${\mathfrak{g}}$ and its Langlands dual ${\mathfrak{g}}^\vee$?
• RQ4 Does the Dynkin index of the principal $\mathfrak{sl}_2$-subalgebra remain invariant under the unfolding of multiply-laced Dynkin diagrams?

Key findings

• The Dynkin index of a principal $\mathfrak{sl}_2$-subalgebra in ${\mathfrak{g}}$ is given by $\frac{\dim{\mathfrak{g}}}{6}h^*({\mathfrak{g}}^\vee)r$, where $h^*({\mathfrak{g}}^\vee)$ is the dual Coxeter number of the Langlands dual ${\mathfrak{g}}^\vee$ and $r$ is the rank of ${\mathfrak{g}}$.
• The index is invariant under the unfolding procedure that maps ${\mathfrak{g}}$ to its simply-laced cover $\tilde{\mathfrak{g}}$, such as $\mathbf{B}_n \to \mathbf{D}_{n+1}$ or $\mathbf{F}_4 \to \mathbf{E}_6$, due to multiplicativity and the fact that ${\mathsf{ind}}({\mathfrak{g}}\hookrightarrow\tilde{\mathfrak{g}}) = 1$.
• The Dynkin index of any irreducible ${\mathfrak{g}}$-module $V_\lambda$ as a $\mathfrak{sl}_2$-module is $\frac{\dim V_\lambda}{6} h^*({\mathfrak{g}}^\vee) r (\lambda, \lambda + 2\rho)_{\mathfrak{g}}$, derived via multiplicativity of indices.
• A new identity is established: $\sum_{i=1}^n \binom{2m_i + 2}{3} = 2h^*({\mathfrak{g}}) \cdot \frac{\dim{\mathfrak{g}}}{6} h^*({\mathfrak{g}}^\vee)r$, where $m_i$ are the exponents of ${\mathfrak{g}}$, linking exponents to dual Coxeter numbers.
• The formula for the sum of squared weights $\sum_{\mu \dashv V_\lambda} (\mu, \rho^\vee)^2_{\mathfrak{g}} = \frac{\dim V_\lambda}{12} h^*({\mathfrak{g}}^\vee) r (\lambda, \lambda + 2\rho)_{\mathfrak{g}}$ is derived and shown to be consistent with the Freudenthal–de Vries formula in the simply-laced case.
• The result provides a uniform expression for $\sum_{\gamma > 0} {\mathsf{ht}}^2(\gamma)$, which is $\frac{\dim{\mathfrak{g}}}{6} h^*({\mathfrak{g}}^\vee)r$, and suggests that such uniform expressions for $\sum {\mathsf{ht}}^s(\gamma)$ may exist only for specific $s$, such as $s=2$.