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51  Foldings of Dynkin diagrams

Remark 51.1 (Orientation) This section collects the folding constructions that produce non-simply-laced root systems from simply-laced ones by summing the roots in each orbit of a diagram automorphism, together with the specific root-folding criterion governing the involution \(I = -\Ien\) on \(\tdp\) and the mirror moves used to navigate the pyramid of \(2\)-elementary lattices. As in the root systems material we work in the algebraic-geometry sign convention: root lattices are negative definite and roots have norm \(-2\) (see Definition 36.2).

51.1 Foldings of simply-laced diagrams

Remark 51.2 (Classical foldings of Dynkin diagrams) The process of folding by nontrivial diagram automorphisms produces non-simply-laced root systems from simply-laced diagrams. The roots in the same orbit under the folding group must be orthogonal.

Classical examples of foldings of simply-laced Dynkin diagrams include:

  • \(A_{2n-1} \to C_n\) via \(S_2\) (horizontal reflection);

  • \(D_{n+1} \to B_n\) via \(S_2\) (vertical reflection);

  • \(D_4 \to G_2\) via \(S_3\) (rotation by \(2\pi/3\));

  • \(E_6 \to F_4\) via \(S_2\) (horizontal reflection).

These arise as the fixed-point subalgebras \(\mathfrak{g}^\sigma\). The orbit-sum construction of Definition 51.3 instead produces the Langlands-dual root system in types \(B\)/\(C\); accordingly the explicit foldings computed below yield \(B_n\) from \(A_{2n-1}\) and \(C_n\) from \(D_{n+1}\) (types \(F_4\) and \(G_2\) are self-dual and unaffected).

51.2 Folded roots and folded root systems

Definition 51.3 (Folded root systems) Let \(L\) be a lattice containing a root system \(\Phi\) and \(G \subset \Orth(L)\) a finite group preserving \(\Phi\). Given a simple root \(\alpha_i \in \Phi\), let \([\alpha_i] \subset \Phi\) denote its \(G\)-orbit.

The associated folded root and folded root system are defined by \[ \beta_{[\alpha_i]} \definedas \Sum_{\alpha_j \in [\alpha_i]} \alpha_j \in L^G, \qquad \Phi^G \definedas \theset{ \beta_{[\alpha_i]} \in L^G \mid [\alpha_i] \in \Phi/G } . \]

Each folded root corresponds to a folded node in the Dynkin diagram. For an involution \(I\), \(\beta_{[\alpha_i]} = \alpha_i + I(\alpha_i) \in L^G\).

51.3 Examples

Example 51.4 (Examples of classical foldings) Folding produces scaled root systems in the invariant lattice \(L^G\). Explicit examples computed via Definition 51.3:

  • \(A_5 \to B_3(2)\): under horizontal reflection (\(G = S_2\)), \(\Phi(A_5^G)\) has Gram matrix \(G_{B_3(2)} = 2 \cdot G_{B_3} = \begin{pmatrix} 4 & -2 & 0 \\ -2 & 4 & -2 \\ 0 & -2 & 2 \end{pmatrix}\) (norms \(4, 4, 2\): two long, one short).

  • \(D_4 \to C_3\): under vertical reflection (\(G = S_2\)), \(\Phi(D_4^G)\) has Gram matrix equal to \(G_{C_3}\).

  • \(D_4 \to G_2\): under rotation by \(2\pi/3\) (\(G = \ZZ/3\ZZ\)), \(\Phi(D_4^G)\) has Gram matrix equal to \(G_{G_2}\).

  • \(E_6 \to F_4(2)\): under horizontal reflection (\(G = S_2\)), \(\Phi(E_6^G)\) has Gram matrix equal to \(G_{F_4(2)}\).

Here the scaling notation \(B_3(2)\) and \(F_4(2)\) records that the invariant-lattice Gram matrix equals the corresponding root-system Gram matrix rescaled by \(2\). (These Gram matrices are written in the standard positive-definite normalization; negate for the AG convention of Definition 36.2.)

51.4 The root-folding criterion for \(\tdp\)

The criterion below is stated in terms of the short and long roots of \(\tdp\): \(\Phi^2\) denotes the short roots (norm \(-2\)) and \(\Phi^4\) the long roots (norm \(-4\) with divisor \(2\)), in the \(k\)-root sense of the root systems material.

Lemma 51.5 (Root folding criterion)

Let \(\Phi(\tdp)\) be the root system of \(\tdp\) and \(I = -\Ien\) the induced involution on \(\tdp\) whose fixed lattice is \(\ten\).

The folded roots \(\beta_{[v]} \in \Phi\left( \tdp^{\generators{I}} \right)\) arise in exactly one of the following ways:

  1. \(v \in \Phi^2(\tdp)\) and \(\beta_{[v]} \in \Phi(\ten)\);

  2. \(v \in \Phi^4(\tdp)\) and \(\beta_{[v]} \in \Phi(\ten)\); or

  3. \(v \in \Phi^2(\tdp) \intersect I(v)^{\perp \tdp}\), and so \(\beta_{[v]} \definedas v + I(v) \in \Phi^4(\ten)\) is the sum of orthogonal roots in \(\Phi^2(\tdp)\).

This criterion governs how roots in the boundary lattices at the \(0\)-cusps of \(\fttz\) descend or combine when passing to the \(I\)-invariant sublattices corresponding to \(\fentwo\).

51.5 Mirror moves

Definition 51.6 (Mirror moves) A mirror move is a lattice-theoretic operation governed by the existence of a primitive isotropic vector \(\eta \in T\) of a specified type and splitting:

  • Odd/simple: \(\div_T(\eta) = 1\), splitting as \(T \cong U \oplus K\). The new invariants are \((r-2, a, 1)\).

  • Even, ordinary: \(\div_T(\eta) = 2\) and \(\eta^*\) is ordinary, splitting as \(T \cong U(2) \oplus K\). The new invariants are \((r-2, a-2, 1)\).

  • Even, characteristic: \(\div_T(\eta) = 2\) and \(\eta^*\) is characteristic, splitting as \(T \cong I_{1,1}(2) \oplus K\). The new invariants are \((r-2, a-2, 0)\).

The move replaces \(T\) with \(\overline{T}_\eta = \eta^{\perp T}/\eta\), navigating Nikulin’s pyramid of \(2\)-elementary lattices to compute cusp diagrams when \(\Gamma = \Orth^+(T)\).