Dynamical Duality-Sequences

• Dynamical duality-sequences are collections of evolving algebraic and analytic structures defined by reciprocal transformation rules between parallel and sequential perspectives.
• They are constructed using techniques like sequentialization, symmetry conjugation, and recursive Bellman equations to enable efficient computations and spectral analyses.
• Their applications span combinatorics, quantum mechanics, and optimal transport, revealing patterns of periodicity, reversibility, and integrability in complex systems.

Dynamical duality-sequences are collections of algebraic, combinatorial, or analytic structures—often evolving in discrete or continuous time—whose defining property is that the evolution, symmetry, or transformation rules are governed by a duality operation. Duality here refers to an exchange between parallel and sequential perspectives, particles and time, operators and their duals, or other conjugate constructs that yield symmetric or recursively interrelated behaviors. Dynamical duality-sequences arise in combinatorics, linear algebra, quantum mechanics, discrete systems, representation theory, and stochastic processes, among other domains. They exhibit complex periodic, hierarchical, or recursive structures, often with implications for optimality, symmetry classification, integrability, spectral properties, or computational efficiency.

1. Parallel-Sequential Duality and Sequential Construction

The foundational framework for dynamical duality-sequences, as articulated in (0709.4397), centers on the parallel-sequential duality for matrices and directed graphs. Matrices, conventionally understood as parallel mappings via

$M^\#(X) = (M_1 \cdot X, \ldots, M_n \cdot X)$

can be reinterpreted as sequential programs. The sequentialization operator $M^\downarrow$ acts via in-place assignments: $$x_i := \sum_{j=1}^n M_{i,j} x_j \quad \text{for } i = 1, \ldots, n$$ This sequential perspective is extended to graphs: the adjacency matrix $M$ of a reflexive directed graph specifies both a parallel connectivity (as a graph) and a sequential constructor (as a program building another graph), resulting in the definition of sequential construction: $$G \text{ sequentially constructs } G' \iff M^\downarrow = M'$$ Iterating the sequential-construction operator $\mathcal{S}(M) = M^\downarrow$ produces dynamical duality-sequences: $$M_0,\, M_1 = \mathcal{S}(M_0),\, M_2 = \mathcal{S}(M_1),\, \ldots$$ Reflexive graphs yield periodic sequences due to the finiteness of regular matrices over finite fields. The duality is reversible up to diagonal correction.

This framework has direct implications for memory-efficient computation (in-place assignment chains with optimal memory), modular decomposition (simplification of graphs/modules up to sequential equivalence), and the emergence of cycles and fixed points in the dynamical system defined by the iterated sequentialization.

2. Quantum Duality-Sequences: Symmetry Conjugates and Operator Dynamics

In quantum mechanics, dynamical duality-sequences manifest through symmetry conjugation and operator duality, as in the Quantum Rabi Model (Omolo, 2021). Here, parity and duality symmetry operators form a closed $SU(2)$ algebra: $$\Pi_j = \sigma_j P,\quad j = x,y,z;\quad [\Pi_j, \Pi_k] = 2i\varepsilon_{jkl}\Pi_l$$ These operators generate duality transformations relating Jaynes-Cummings and anti-Jaynes-Cummings Hamiltonians, with duality conjugation mapping: $$\Pi_y H_{I,JC} \Pi_y = H_{I,aJC}, \quad \Pi_x H_{I,JC} \Pi_x = -H_{I,aJC}$$ Linear combinations of the Hamiltonian and its dual yield three exactly solvable dynamical regimes (bosonic, fermionic, coupling-only), each describing distinct physical evolutions (entangled cat states, spin coherent states, etc.), organized in a duality-sequence structured by symmetry group algebra.

The concept generalizes to other quantum models with similar algebraic structures, enabling systematic exploration of dynamical regimes via chains of duality transformations.

3. Particle-Time Duality and Trace Identities in Spin Chains

In discrete quantum spin systems such as the kicked Ising chain (Akila et al., 2016), dynamical duality-sequences arise as a symmetry between evolution in space and time. The trace of the Floquet operator for $N$ spins evolved for $T$ steps,

$\operatorname{Tr} U_N^T$

is exactly equal to the trace of a dual non-unitary operator (dimension $2^T$) evolved for $N$ steps: $$\operatorname{Tr} U_N^T = \operatorname{Tr} \tilde{U}_T^N$$ The dual operator $\tilde{U}_T$ is constructed with dualized interaction parameters and encapsulates the spectral and dynamical features of the original system, allowing the paper of spectral statistics, chaos, and regularity via cyclic evolution in the dual direction. The duality-sequence thus represents a temporal-particle interchange, underpinning universality and computational efficiency in analyzing spin-chain dynamics.

4. Dynamical Duality-Sequences in Recurrence and Spectral Theories

Combinatorial duality-sequences govern the behavior of recurrence relations in algebraic geometry and singularity theory, notably in the duality defect conjecture (Jorgenson, 2018). The degree sequence of Segre classes,

$$s_0 = 1, \quad s_j = \sum_{q=1}^{\min(j,m)} (-1)^{q+1} c_q s_{j-q}$$

acts as a dynamical duality-sequence whose positivity and pattern of vanishing encode the geometric property of duality defect in projective varieties. The dynamical aspect arises from the evolution and constraints on the sequence as parameters and dimension vary, with exhaustive algorithmic bounds indicating where counterexamples may or may not occur. The possibility or impossibility of certain sequences (positivity before and zeros at specified positions) determines the existence of defect varieties, linking combinatorics, algebraic geometry, and number theory into a duality-driven dynamical process.

5. Stochastic, Algebraic, and Representation-Theoretic Duality-Sequences

In stochastic integrable systems, duality-sequences involve hierarchies of duality functions (orthogonal polynomials) connecting interacting particle processes (Wagenaar, 18 Feb 2025, Borodin et al., 2017). For dynamic ASIP, dualities are mediated by Askey-Wilson, Al-Salam--Chihara, and Jacobi polynomials, organized into a hierarchical Askey-scheme. These polynomials appear as matrix elements of quantum algebra representations, and the nested structure ("orthogonal" vs. "triangular" duality) reflects limits and degenerations among the processes, with each transition corresponding to a change in the duality function.

Similarly, in representation theory (Dalipi et al., 2022, Bodish et al., 13 May 2024, Tarasov et al., 2019), dynamical duality-sequences organize operator actions under Schur–Weyl or Howe duality: collections of KZ/dynamical operators, dynamical Weyl group actions, or fusion operators generate interrelated symmetric structures on module categories, often with explicit braid and reflection relations. Sequences of such operators—differential, difference, or fusion constructs—encode the algebraic interrelations and flatness/monodromy properties of connections across dual pairs (e.g., $(\mathfrak{gl}_N, \mathfrak{gl}_M)$ or split symmetric pairs $(\mathfrak{so}_{2n}, O_m)$).

6. Dynamical Duality-Sequences in Discrete Dynamical Systems and Data Processing

The algebraic duality of sequence and function spaces yields dynamical duality-sequences in discrete linear system theory [(Andriamifidisoa et al., 2012); (Aandriamifidisoa et al., 15 Jul 2025)]. Here, duality sequences link finite-support and arbitrary (possibly two-sided) indexed sequences, with adjointness relations between polynomial multiplication and shift/convolution operators,

$$d(X): c(X) \mapsto c(X)d(X), \qquad d(X) \circ W(Y) = \sum_{a, \beta} d_a W_\beta Y^{a+\beta}$$

Bidirectional discrete systems—expressible via Laurent series—exhibit sequences of states or filters whose evolution and transformation are governed by duality between the signal and operator spaces, with implications for multidimensional data processing, signal filtering, and control theory.

7. Duality-Sequences in Dynamical Optimal Transport

In control and optimal transport theory, the dual formulation of discrete-time dynamical optimal transport (Wu et al., 13 Oct 2024) is naturally expressed as a sequence of time-indexed value functions $\{v_k\}$, linked by recursive Bellman equations: $$v_k(x) = \inf_u\left\{L_k(x,u) + v_{k+1}(f_k(x,u))\right\}$$ The optimization is governed not by a static Kantorovich potential pair, but by a dynamical duality-sequence—a chain of value functions encoding the evolution, constraints, and optimal behavior of the transport process. The sequence is both the dual certificate and the source of optimal control policies; its structure permits efficient, equation-free algorithms via first-order splitting and parallelization.

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Summary Table: Examples of Dynamical Duality-Sequences

Domain	Objects	Duality-Sequence Manifestation
Linear algebra/combinatorics	Matrices/graphs	Sequentialization, iterated sequential constructors ($M^\downarrow$)
Quantum mechanics	Hamiltonians/operators	Chains of symmetry conjugates, SU(2) algebra, dynamical regimes
Spin chains/statistical mechanics	Evolution operators	Particle-time interchange, dual trace identities
Algebraic geometry	Recurrence sequences	Positivity/vanishing patterns encode geometric invariants
Stochastic processes	Duality functions	Hierarchies of orthogonal polynomials, degeneration limits
Representation theory	Weyl/fusion/group operators	Sequences of dynamical actions, braid and reflection relations
Control/data processing	Sequence spaces	Duality between signals and operators; bidirectional filtering and convolution
Optimal transport	Bellman value functions	Time-indexed dual potentials define transport/control policy sequences

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Dynamical duality-sequences represent a unifying principle across mathematics, physics, and data sciences. Their explicit construction via duality operations reveals deep interconnections between symmetry, recursion, evolution, and optimality. The periodicity, reversibility, and hierarchical structure observed in such sequences underpin a wide range of phenomena from combinatorial dynamics, quantum integrability, stochastic duality, to control and optimization, providing rigorous tools for both structural analysis and computational approaches.