Dynkin Z₂ Swap: Involutive Symmetry Action

• Dynkin Z₂ Swap is a discrete symmetry that involutively exchanges paired elements in algebraic structures such as Dynkin and Carter diagrams.
• It underpins key transformations in Lie superalgebra diagrams by swapping bosonic and fermionic roots, thereby preserving diagrammatic properties.
• The swap also appears in incidence algebras and particle physics models, offering insights into derived equivalence and universal mass ratio predictions.

The Dynkin $\mathbb{Z}_2$ Swap is a discrete symmetry action, typically realized as an involutive automorphism or transition, which exchanges certain structural components in algebraic or diagrammatic representations associated with Dynkin diagrams, root systems, Lie superalgebras, incidence algebras, and related frameworks. Its canonical appearance is as a two-element symmetry (order-two map) that effects a transformation—most commonly a "flip" or "swap"—between paired objects, such as bosonic and fermionic roots in superalgebra Dynkin diagrams, root basis vectors in Carter diagrams, or algebra presentations up to isomorphism with the opposite algebra. The concept is pervasive in representation theory, algebraic combinatorics, and physical models where $\mathbb{Z}_2$-graded structures or symmetry breaking play a critical organizing role.

1. Involutive Symmetries in Dynkin and Carter Diagrams

The Dynkin $\mathbb{Z}_2$ Swap arises explicitly as an involutive transition among root subsets associated to Carter diagrams of the same $ADE$ type and cardinality (Stekolshchik, 2022). Given two bases $\tilde{\Omega}$ and $\Omega$ differing by exactly one root, a linear mapping $M_i$ is defined such that

$$\Omega = \{\tau_1, \ldots, \tau_{i-1}, \alpha, \tau_{i+1}, \ldots, \tau_n\}, \quad \alpha = -\tilde{\alpha} + \sum_{j\ne i} t_j \tau_j$$

with integer coefficients $t_j$ selected for orthogonality. The transition matrix $M_i$ is involutive ($M_i^2 = I$), interchanging $\tilde{\alpha}$ and $\alpha$. This $\mathbb{Z}_2$ symmetry exchanges root bases within the Weyl group conjugacy class, preserving diagrammatic structure.

Moreover, Carter diagrams and enhanced Dynkin diagrams (especially $\Delta(E_6)$, $\Delta(E_7)$, $\Delta(E_8)$) exhibit a relationship wherein the number of non-Coxeter Carter diagrams and the number of extra nodes added in the enhancement process both match familiar $2-4-8$ patterns, connecting this involutive swap to wider algebraic phenomena (Stekolshchik, 2022).

2. The Flip in Dynkin Superdiagrams of Lie Superalgebras

In the setting of Lie superalgebras, the Dynkin $\mathbb{Z}_2$ Swap finds direct instantiation as the "flip" map $f$ between bosonic and fermionic roots in Dynkin diagrams (Ransingh, 2013). The construction leverages boson–fermion correspondence and $Z_2$-grading ($$\mathfrak{g} = \mathfrak{g}_0 \oplus \mathfrak{g}_1$$), mapping

$$f : \{\text{bosonic roots}\} \leftrightarrow \{\text{fermionic roots}\}$$

within the diagram. In simply laced diagrams, this preserves the adjacency and orientation structure; white nodes (bosonic) are swapped with grey/black nodes (fermionic) and vice versa. This operation yields the flip Dynkin superdiagram, and induces a corresponding flip in Cartan matrices (with parity-sensitive entries). Crucially, the defining relations and Serre-type relations are similarly swapped, using the new Cartan matrix structure.

This $\mathbb{Z}_2$ operation creates non-conjugate classes of Borel subalgebras and non-isomorphic Dynkin diagrams within the same algebra, showcasing the richness introduced by the grading and swap.

3. $\mathbb{Z}_2$ Actions in Piecewise Hereditary Algebras and Incidence Algebras

In the theory of piecewise hereditary (PHI) algebras and incidence algebras of Dynkin/extended Dynkin type, the Dynkin $\mathbb{Z}_2$ Swap manifests as the symmetry between an algebra and its opposite algebra, particularly in the classification theorems (Marcos et al., 2017). For a given incidence algebra $K\Delta$ or its presentation $KQ/I$, the derived equivalence class is invariant under orientation reversal: $T(A)^{op} \cong T(A^{op})$ where $T(A)$ is the trivial extension. The operation swaps all arrows, corresponding to a $\mathbb{Z}_2$ action. This symmetry is reflected in computational procedures, such as the selection of cutting sets in the trivial extension quiver, where different choices yield algebras related by the swap. Classification results frequently state isomorphism "up to the opposite"—this ambiguity is the $\mathbb{Z}_2$ swap.

The swap is central to the determination of PHI algebras and their derived categories, emphasizing that algebraic properties are "swappable" under this involutive symmetry.

4. The Role of the Dynkin Swap in Octonionic and Jordan Algebra Models

Within particle physics models based on exceptional Jordan algebra $J_3(\mathbb{O}_\mathbb{C})$ and the symmetric cube representation $\mathrm{Sym}^3(\mathbf{3})$ of $\mathrm{SU}(3)$, the Dynkin $\mathbb{Z}_2$ Swap again appears as the automorphism (A$_2$ diagram flip) acting on ladder structures for fermion masses (Singh, 13 Aug 2025). The symmetric cube ladder is organized by Clebsch weights (2,1,1) and the swap $S$ exchanges two SU(3) directions while fixing the anchor: $$S: (a, b, c) \longrightarrow (a, c, b), \quad S^2 = I$$ This operation transforms the down-quark ladder into the charged-lepton ladder, enforcing relations such as

$$\sqrt{\frac{m_\tau}{m_\mu}} = \sqrt{\frac{m_s}{m_d}}$$

By "flipping" one leg of the ladder, the swap guarantees the universality of adjacent mass-ratio predictions across sectors, connected to the underlying Jordan spectrum. Furthermore, in mixing (CKM) parameters, the swap plays a crucial role in organizing geometric relations.

5. $\mathbb{Z}_2$ Symmetry Operations in Physical Models

A broad class of physics models, especially those with discrete symmetries in scalar sectors—such as the Inert Doublet Model (IDM) and Two Real Singlet Model (TRSM)—exhibit $\mathbb{Z}_2$ symmetry and its breaking in their phenomenological structure (Robens, 2022). While the direct link to Dynkin diagram swaps is not explicit, the underlying mathematical ethos—using discrete involutive operations to relate sectors, eigenstates, or decay patterns—mirrors the logic of Dynkin $\mathbb{Z}_2$ swaps. The mapping of mass splittings, mixing patterns, and phenomenological signatures may be interpreted as physical manifestations of the same algebraic symmetry principle.

This points toward a unified framework wherein discrete swaps organize both mathematical classification and physical predictions, with possible implications for future model-building and experimental search strategies.

6. Mathematical Properties and Generalizations

The Dynkin $\mathbb{Z}_2$ Swap generally satisfies:

• Involutivity: $S^2 = \mathrm{id}$ or $M_i^2 = I$, interchange of paired elements.
• Commutation with Symmetry Group: Transition matrices $M_i$ commute with Weyl group actions: $w M_i = M_i w \;\forall w \in W$ (Stekolshchik, 2022).
• Invariant Substructure: Transformation rules preserve diagrammatic or algebraic classes up to the swap; Cartan matrices transform via $M_i^T B_{\tilde{\Gamma}} M_i = B_\Gamma$.
• Numerical Patterns: The $2-4-8$ arithmetic in enhanced Dynkin diagrams corresponds to $\mathbb{Z}_2$-classified objects.

These properties render the swap a fundamental symmetry tool in the classification of algebraic and combinatorial structures, with pervasive generalizations across representation theory, superalgebra, and physical model construction.

7. Significance and Interconnections

The Dynkin $\mathbb{Z}_2$ Swap serves as the paradigm for discrete symmetry operations that mediate fundamental equivalences, dualities, and observable relations within algebraic and physical settings. Its presence is emblematic of deeper coherence between geometric, combinatorial, and physical hierarchies—whether as the flip of bosonic/fermionic roots in superalgebras, the swap of root subsets in Carter diagrams, the equivalence of algebra/opposite algebra structures, or the transfer of physical ratios and mixing parameters in exceptional algebra models. The general principle enables a systematic treatment of structural diversity (non-isomorphic representations, alternative Borel subalgebras, universal mass ratios), and highlights the centrality of involutive symmetries in both mathematics and theoretical physics.