Dual Root Systems
The dual root system is the set of coroots associated to a root system. It is itself a root system in the dual vector space, and the operation \(\Phi \mapsto \Phi^\vee\) is an involution that interchanges \(B_n\) with \(C_n\) and is the identity (with long-short swap) on non-simply-laced types \(F_4, G_2\).
Definition
For a root system \(\Phi \subseteq V\) (a Euclidean space with inner product \((-, -)\)), the dual root system is \[ \Phi^\vee \;=\; \{\alpha^\vee : \alpha \in \Phi\} \;\subseteq\; V, \] where \(\alpha^\vee = 2\alpha / (\alpha, \alpha)\) is the coroot of \(\alpha\).
\(\Phi^\vee\) is a root system in \(V\), with the same Weyl group as \(\Phi\). The bijection \(\Phi \leftrightarrow \Phi^\vee\), \(\alpha \leftrightarrow \alpha^\vee\), is a canonical bijection.
Properties
- \((\Phi^\vee)^\vee = \Phi\) (involution).
- \(W(\Phi) = W(\Phi^\vee)\) as a subgroup of \(\mathrm{GL}(V)\).
- The Cartan matrix \((c_{ij})\) of \(\Phi\) transposes to \((c_{ij}^\vee) = (c_{ji})\) for \(\Phi^\vee\) (with appropriate base correspondence).
Effect on Dynkin type
For irreducible root systems:
| \(\Phi\) | \(\Phi^\vee\) |
|---|---|
| \(A_n\) | \(A_n\) (self-dual) |
| \(B_n\) | \(C_n\) |
| \(C_n\) | \(B_n\) |
| \(D_n\) | \(D_n\) (self-dual) |
| \(E_6, E_7, E_8\) | self-dual |
| \(F_4\) | \(F_4\) (long-short swap) |
| \(G_2\) | \(G_2\) (long-short swap) |
For simply-laced types (\(A, D, E\)): \(\Phi = \Phi^\vee\) literally (no length distinction).
For non-simply-laced types: \(\Phi^\vee\) is the root system obtained by scaling each root by \(2/(\alpha, \alpha)\), which inverts the length ratio.
Cartan matrix transposition
If \(C = (c_{ij})\) is the Cartan matrix of \(\Phi\) (for a chosen base), then the Cartan matrix of \(\Phi^\vee\) is the transpose \(C^T = (c_{ji})\). This swap exchanges:
- The role of "row" (= \(\alpha_i\)) and "column" (= \(\alpha_j^\vee\));
- The arrow direction in non-simply-laced Dynkin diagrams.
Examples
\(A_n\) (self-dual)
All roots have the same length, so \(\alpha^\vee = \alpha\) (up to normalisation). \(\Phi^\vee\) is literally the same set as \(\Phi\).
\(B_2\) and \(C_2\)
\(B_2\): short roots \(\pm \varepsilon_i\), long roots \(\pm \varepsilon_i \pm \varepsilon_j\). Coroots scale short to long and vice versa, giving \(C_2\).
\(G_2\)
Long roots scaled by \(1/3\) become short coroots; short roots scaled by 1 stay short. Result: the dual root system is \(G_2\) with long and short roles swapped.
Connection to root data
A root datum is a root system enriched with a lattice \(\mathbb{Z}\Phi \subseteq X \subseteq P\). Under root-datum duality (swapping characters and cocharacters), the underlying root system maps to its dual root system, and the lattice \(X\) maps to a specific dual lattice in \(V^*\).
The dual of a root datum is not just the dual of the underlying root system — it also inverts the lattice gap (sc ↔ ad).
Geometric interpretation
In the geometric setting:
- \(\Phi\) = weights of \(T\) acting on \(\operatorname{Lie}(G)\);
- \(\Phi^\vee\) = cocharacters arising from \(\mathfrak{sl}_2\)-triples.
For the Langlands dual: roots and coroots swap. This is the combinatorial source of all Langlands duality phenomena.
Mathlib
Mathlib.LinearAlgebra.RootSystem.RootPairing formalises root
pairings (= root systems with their dual data). The dual operation is
built in.
References
- Bourbaki, Groupes et Algèbres de Lie IV–VI (1968).
- Humphreys, Introduction to Lie Algebras and Representation Theory (1972).