Harder–Narasimhan Filtrations Overview

• Harder–Narasimhan Filtrations are canonical decompositions that split objects into semistable pieces with strictly decreasing slopes across various mathematical categories.
• They are generalized by order-theoretic, categorical, and metric-geometric frameworks to unify stability concepts in algebraic geometry, representation theory, and beyond.
• Their applications span moduli theory, GIT stratifications, tensor product compatibility, and efficient algorithmic computation in representation theory.

A Harder–Narasimhan (HN) filtration is an intrinsically defined, canonical filtration of an object in a suitable category (e.g., vector bundles, modules, quiver representations, moduli-theoretic stacks), whose graded pieces are "semistable" with respect to an explicit slope function, and whose slopes strictly decrease. Originally found in algebraic geometry for vector bundles on curves, the theory has been abstracted and unified across domains by categorical, order-theoretic, and metric-geometric formalisms. The HN filtration exists uniquely for a broad class of objects and provides deep structural invariants, as well as a bridge to moduli theory, representation theory, and arithmetic geometry.

1. General Framework and Game-Theoretic Formalism

The foundational modern approach, due to Chen–Jeannin, recasts the HN formalism in purely order-theoretic terms (Chen et al., 2023). Given a bounded lattice $\mathscr{L}$ of subobjects (with top $\top$ and bottom $\perp$) and a totally ordered complete lattice $S$, the "payoff function" $\mu : P_<(\mathscr{L}) \to S$ associates a "slope" to every proper interval $(x, y)$ with $x < y$. The central construction is a two-player zero-sum sequential game, where Alice and Bob pick subobjects (with Alice minimizing and Bob maximizing $\mu$), yielding min-sup and sup-inf payoffs $\mu_A^*,\ \mu_B^*$. Semistability is formalized as the condition $\mu_A(y) \leq \mu_A(\top)$ for all $y \neq \perp$, directly paralleling "no subobject has slope strictly greater than the whole".

The abstract existence and uniqueness of the Harder–Narasimhan filtration are guaranteed under:

• Convexity of $\mu$, i.e., $\mu(x \wedge y, x) \leq \mu(y, x \vee y)$,
• Ascending chain condition (ACC) and a mild $\mu_A$-descending-chain condition on $\mathscr{L}$,
• $S$ totally ordered (or infima attained).

The canonical filtration is a chain $\perp = a_0 < a_1 < \cdots < a_n = \top$ such that on each interval $[a_{i-1}, a_i]$ the game is semistable and the thresholds $\mu_A(a_{i-1}, a_i)$ strictly decrease.

The entire classical HN formalism for vector bundles, Gieseker stability for sheaves, quiver representations, $p$-adic Hodge objects, and even coprimary filtrations for Noetherian modules is recovered by selecting the appropriate $\mathscr{L}$ and $\mu$ (Chen et al., 2023, Yuan, 23 Sep 2025).

2. Classical Cases and Categorical Formulations

The standard algebraic geometric setting is that of an abelian (or proto-abelian) category $\mathcal{C}$ with additive degree and rank functions. The slope is defined as $\mu(E) = \deg(E)/\mathrm{rk}(E)$. An object $E$ is (semi)stable if every nonzero subobject $F$ satisfies $\mu(F) \leq \mu(E)$.

Categorified Viewpoint: The theory has been extended to categories lacking additive decompositions via the slope function formalism (Li, 2021). Given a (partial) order $\Lambda$ of slopes and a function $\mu$ defined on subquotients, the existence and uniqueness of the HN filtration is established via order-theoretic arguments, leveraging monotonicity and categorical pushout/pullback structure rather than short exact sequences.

This axis of abstraction encompasses vector bundles, normed lattices, filtered modules, Euclidean lattices, and even linear codes (Li, 2021); the only requirement is a sufficiently well-behaved notion of subobject and a monotone slope function.

3. Metric, Building, and Tensor Product Compatibility

Recent advances have exploited the metric and convex geometry structure on spaces of filtrations. Building on modular lattice theory, Lagier and other contributors attached "buildings" to the set of $\mathbb{R}$-filtrations, with a CAT(0) metric and a scalar product encoding the rank structure (Cornut, 2017).

Key Result: In such a setup, the HN filtration uniquely minimizes a "distance-to-origin minus degree functional." This geometric framework yields an abstract numerical criterion for compatibility with tensor products: the HN filtration commutes with tensor product if for all $E$, $F$,

$$\mu_{\max}(E \otimes F) \leq \mu_{\max}(E) + \mu_{\max}(F).$$

Convex projection in the CAT(0) building yields the unique HN-filtration, and Busemann pairings ensure formal compatibility. This has enabled axiomatization of HN-filtrations in highly general tensor categories, including normed $\varphi$-modules and shtukas, with links to $p$-adic Hodge theory and the Langlands program (Cornut, 2017, Cornut et al., 2018).

4. Algorithmic and Representation-Theoretic Aspects

In the theory of quiver representations and moduli, the HN filtration coincides with the unique filtration measuring the failure of slope-semistability, and underlies both geometric invariant theory (GIT) stratifications and the construction of moduli stacks (Zamora, 2012, Cheng, 2021). Notably, deterministic polynomial-time algorithms now exist for computing HN-filtrations in quiver representations over infinite fields, leveraging subrepresentation discrepancy functions and links to maximally destabilizing 1-parameter subgroups in GIT (Cheng, 2021).

Table: Algorithmic Aspects for Quiver Representations

Main Operation	Methodology	Complexity
Compute HN-filtration	Discrepancy subroutine, maximal subrep	Polynomial in dimension, weights, #paths (Cheng, 2021)
Max. destabilizing 1-PS	Kempf's theorem, GIT refinement	Polynomial for bipartite quivers (Cheng, 2021)

The HN filtration stratifies representation spaces, the nullcone of the moment map, and Nakajima quiver varieties, with codimension formulas explicating their link to Kac polynomials and equivariant cohomology (Yau, 4 Nov 2025).

5. Moduli-Theoretic, Stratification, and Derived-Stack Perspectives

The existence and uniqueness of HN filtrations drive stratification phenomena in moduli theory. For families of principal $G$-bundles, the relative HN filtration stratifies the parameter scheme into locally closed subschemes with a universal property: the fiber over any base change corresponds precisely to those objects admitting a global relative canonical reduction of the given HN-type (Gurjar et al., 2012). These stratifications descend to Artin stacks, and their combinatorics control both the geometry and deformation theory of moduli (Núñez, 2023).

In derived algebraic geometry, explicit derived moduli stacks of HN filtrations have been constructed, with tangent complexes controlled by derived endomorphisms of the filtered object, and universal properties extending classical moduli—the representing derived 1-stack enhances the classical Artin stratum and encodes deformation/obstruction theory (Mizuno, 2022).

6. Interplay with Geometric Invariant Theory

A central discovery is the identification of HN filtrations with "maximally destabilizing" 1-parameter subgroups in GIT. For unstable objects in moduli problems (sheaves, quiver representations, Higgs bundles, decorated sheaves), the Kempf–Rousseau theorem asserts that the maximally destabilizing 1-PS induces (through its weight filtration) exactly the HN filtration (Zamora, 2014, Zamora, 2012, Zamora, 2013).

The Hesselink/Kirwan strata for GIT quotients coincide with HN-type stratifications, yielding recursive formulas for moduli cohomology, wall-crossing, and explicit moduli stratification (Zamora, 2014). This bridge is highly robust, holding uniformly for sheaves, principal bundles, and representations.

7. Applications, Obstruction Theory, and Extensions

The HN filtration formalism subsumes, and is used to uniquely characterize:

• Coprimary filtrations of Noetherian modules: the unique filtration with successive coprimary quotients indexed by strictly decreasing associated primes (Chen et al., 2023, Yuan, 23 Sep 2025).
• Canonical reductions of principal bundles (e.g., Atiyah–Bott, Biswas–Holla): explicit procedures construct the canonical reduction to a parabolic, with the HN type describing the "most unstable" reduction (Roth et al., 21 Nov 2025).
• p-adic Hodge theory: for Breuil–Kisin–Fargues modules, there is a unique HN filtration functorial with respect to Tate twists and tensor products, crucial for the integral $p$-adic Hodge landscape (Cornut et al., 2018).
• Stratification of moduli stacks: the hard structure of HN-type stratification persists in the analytic and derived settings (Núñez, 2023, Gurjar et al., 2012, Mizuno, 2022).
• Persistence modules and applied topology: the HN-type and its "skyscraper" invariant rigidly refine the usual rank-invariant in the theory of multiparameter persistence (Fersztand et al., 2023, Fersztand, 2024).

Future directions include metric and convex-geometric generalizations (e.g., building-theoretic formalism), extension to derived moduli, and the categorification of "wall-crossing" and Hall algebra decompositions in both abelian and triangulated frameworks (Gao et al., 25 Dec 2025).

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References: For technical details and development of these themes, see (Chen et al., 2023, Cornut, 2017, Cornut et al., 2018, Li, 2021, Yuan, 23 Sep 2025, Cheng, 2021, Zamora, 2012, Treffinger, 2018, Núñez, 2023, Yau, 4 Nov 2025, Gao et al., 25 Dec 2025, Gurjar et al., 2012, Roth et al., 21 Nov 2025, Mizuno, 2022, Fersztand et al., 2022, Fersztand et al., 2023, Fersztand, 2024), and the cited references therein.