H-Invariance Theory in Mathematics

• H-Invariance Theory is a unified mathematical framework that characterizes preserved structural properties under symmetry operations, algebraic extensions, or dynamical evolutions.
• It provides a complete algebraic and invariant-based analysis in optimal fixed-point algorithms, offering explicit polynomial invariants and certificate conditions for minimax-optimality.
• The theory extends to various fields—including dynamical systems, operator theory, algebraic geometry, and K-theory—enabling precise classification and invariant preservation across disciplines.

H-Invariance Theory encompasses a set of mathematical frameworks unified by the principle that certain structural or quantitative features (“invariants”) are preserved under prescribed symmetry operations, algebraic extensions, or dynamical evolutions. The term “H-invariance” appears across several subfields—including dynamical systems, operator theory, algebraic geometry, invariant theory, and optimization—each context instantiating specific invariants and the relevant symmetry or transformation group (often denoted by H). H-invariance theory investigates, characterizes, and exploits such invariance, often extracting deep structural classification results, optimality conditions, quantitative asymptotics, or algebraic generators.

1. H-Invariance in Optimization and Fixed-Point Algorithms

In the context of optimal acceleration for fixed-point algorithms, H-invariance theory is developed as a complete algebraic characterization of deterministic first-order methods that achieve exact minimax-optimal worst-case convergence rates on nonexpansive fixed-point problems.

Algorithms in this setting are described by lower-triangular step-size matrices $H$ (“H-matrices”), and their performance is characterized by a finite set of polynomial invariants $P(N-1,m;H)$ (“H-invariants”), associated to the number of steps $N$. Each algorithm’s worst-case performance is fully determined by the values of these invariants, with the optimal set given by $P(N-1,m;H) = \frac{1}{N}\binom{N}{m+1}$ for $m=1,\ldots,N-1$.

The region of actual minimax-optimality within this set is precisely specified by a secondary set of polynomial inequalities—“H-certificates”—which take the form $\lambda^\star_{k,j}(H)\geq 0$, for $1\leq j<k\leq N$. The classical optimal Halpern method, its H-dual, and all other optimal algorithms correspond to H-matrices lying in the intersection of this algebraic level set and the certificate-defined region. This framework exhaustively describes all optimal span-based methods and explains the non-uniqueness of solutions via the geometry of the “certificate cone” within the space of H-matrices (Yoon et al., 18 Nov 2025).

<table> <thead> <tr> <th>Object</th> <th>Definition/Role</th> <th>Optimality Condition</th> </tr> </thead> <tbody> <tr> <td>H-matrix $H$</td> <td>Lower-triangular step-size matrix parametrizing algorithm</td> <td>$P(N-1,m;H)=\frac{1}{N}\binom{N}{m+1}$</td> </tr> <tr> <td>H-invariants $P(N-1,m;H)$</td> <td>Homogeneous polynomials in $H$ entries, determining output</td> <td>Fixed as above</td> </tr> <tr> <td>H-certificates $\lambda^\star_{k,j}(H)$</td> <td>Polynomial inequalities specifying optimality region</td> <td>All $\geq0$</td> </tr> </tbody> </table>

This theory replaces algorithm-specific Lyapunov/proof designs by algorithm-agnostic invariant and certificate analysis, and generalizes to related settings such as nonsmooth minimization (Yoon et al., 18 Nov 2025).

2. H-Invariance for Dynamical Systems: Hölder Invariance Principles

In dynamical systems, “Hölder-invariance” theory refers to functional invariance principles in Hölder spaces for the partial-sum (Birkhoff) process of certain mixing, nonuniformly expanding maps. Within the Gibbs–Markov/Young tower formalism, a key result is that, under a “weak p-moment” condition on the return-time function (specifically, $\mu\{\tau>n\}=O(n^{-p}L(n)^{-1})$ for $p>2$ and $L$ slowly varying), the normalized partial sum process for any Hölder continuous observable $\varphi$ with mean zero exhibits weak convergence in a Hölder path space of index $\theta = 1/2 - 1/p$ to Brownian motion with explicitly computable variance (“Hölder invariance principle”). This extends the classical (vector-valued) Donsker–Prokhorov invariance principle to rough path spaces in the presence of long-range memory and non-uniform expansion (Alouin et al., 19 Feb 2025).

Furthermore, the invariance principle extends to observables of bounded variation (BV), under suitable mixing coefficients for the transfer operator acting on BV. The proof architecture leverages:

• Reduction to an induced Markov shift on the tower.
• Precise transfer of tail bounds for return times.
• Coupling methods (Kantorovich–Rubinstein dual formula) to establish decay of mixing coefficients.
• Application of functional CLTs (e.g., Giraudo, Dedecker–Merlevède) for sequences with $\tau$-mixing or $\alpha$-mixing properties.

This theory is sharp for intermittent maps of Pomeau–Manneville type, including quantitative results on scaling exponents and variance (Alouin et al., 19 Feb 2025).

3. H-Invariance in Algebra, Representation Theory, and Polynomial Identities

In the context of polynomial identity (PI) theory for associative algebras equipped with generalized Hopf algebra actions (i.e., “H-module algebras”), “H-invariance” primarily concerns the invariance of the Jacobson radical $J(A)$ under the action of $H$, and structures arising from this property. The main result is that given a finite-dimensional algebra $A$ over an algebraically closed field of characteristic zero, with a generalized $H$-action and $J(A)$ $H$-invariant, the sequence of “$H$-codimensions” $c_n^H(A)$ grows asymptotically as $d^n$ up to polynomial factors, where $d$ is given in terms of the maximal sum of dimensions of $H$-simple blocks in $A/J(A)$ that interact nontrivially under the $H$-action.

H-invariance thus unifies and extends prior work on graded, differential, and group-acted algebras, and generalizes Amitsur's conjecture for PI-exponents to this non-semisimple and nontrivial-action context. This provides explicit criteria for the exponential rate (PI-exponent) and connects the asymptotics to intrinsic structural invariants determined by $H$-invariance of the radical and module decompositions (Gordienko, 2012).

4. H-Invariance in Invariant Theory and Compact Group Actions

H-invariance arises classically in invariant theory, with focus on polynomial functions invariant under the action of a compact Lie group $G$, containing a normal subgroup $H$ of finite index. The $H$-invariants form an algebra $C[V]^H$ decomposed into a direct sum of $m$ submodules (where $m=[G:H]$), corresponding to distinct “relative invariants” parametrized by a character $\omega^j$. The relative Reynolds operators $R_j$ project onto these submodules, and their explicit decomposition yields a constructive algorithm (via Hilbert basis lifting) to generate the $G$-invariant algebra from the $H$-invariant algebra.

This approach provides both the structural direct-sum theorem and explicit generators for the ring $C[V]^G$ in terms of the ring $C[V]^H$ and the relative Reynolds operators, thereby forming a core of H-invariance theory in algebraic invariant theory (Baptistelli et al., 2012).

5. H-Invariance in Operator Theory: Hyperinvariant Subspaces

In operator theory, the term “H-invariant subspace” is synonymous with “hyperinvariant subspace,” that is, a closed subspace $M$ invariant under every bounded operator commuting with a fixed bounded operator $T$ on a Hilbert (or right quaternionic—“right Hamilton”) space. The celebrated “hyperinvariant subspace problem” was resolved affirmatively: every bounded operator on a separable infinite-dimensional Hilbert (or right Hamilton) space possesses a nontrivial closed hyperinvariant subspace. The proof utilizes compactness and abelian von Neumann algebra constructions derived from the commutant of $T$ (Lee, 2023).

Hyperinvariant subspaces are of critical importance in the structure theory of operators, spectral theory, and noncommutative geometry. Their existence guarantees a higher-level “invariance” than mere $T$-invariance, with direct applications to representation theory and quantum information when considering stability under large symmetry (commutant) groups.

6. H-Invariance in Algebraic Geometry: Derived Categories and Hodge Number Invariance

In algebraic geometry, H-invariance refers to the phenomenon whereby certain numerical invariants—most notably the Hodge numbers $h^{0,p}(X)$—are preserved under derived equivalence of smooth projective varieties. If $X_1$ and $X_2$ are derived equivalent via a Fourier–Mukai transform, then $h^{0,p}(X_1) = h^{0,p}(X_2)$ for all $p$. The proof crucially passes through the invariance of Hochschild homology and the Hochschild–Kostant–Rosenberg decomposition, which canonically projects to the $(0,p)$ Hodge component. This result positions $h^{0,p}$ as central invariants under the “largest reasonable” motivic equivalence furnished by the derived category, placing strong constraints on possible derived equivalences and hinting at deeper Hodge-theoretic relations preserved in the noncommutative context (Abuaf, 2019).

7. H-Invariance in Homotopy Theory and Algebraic K-theory

H-invariance is manifested in the homotopy invariance of higher algebraic K-theory. Here, passage from a noetherian abelian category $\mathcal{A}$ to its “polynomial” extension $\mathcal{A}[t]$ (objects equipped with endomorphisms) induces an isomorphism in higher K-groups: $K_n(\mathcal{A}) \cong K_n(\mathcal{A}[t])$ for all $n\geq 0$. The proof relies on graded category theory, Koszul complexes, and localization techniques. This result immediately specializes to classical $A^1$-homotopy invariance for $K'$-theory and G-theory of schemes, reflecting a robust invariance under (derived) polynomial extension. The result is foundational in the paper of motives and algebraic cycles, with ramifications for the classification of schemes up to “homotopical” equivalence (Mochizuki et al., 2013).

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The concept of H-invariance, in its various forms, underpins a broad class of deep mathematical results where structure, quantitative behavior, or classification is dictated by invariance under group actions, algebraic transformations, or combinatorial constraints. In each context, the precise identification and exploitation of invariants—together with sharp criteria for when invariance leads to optimality, generation, or classification—constitutes the heart of H-invariance theory.