Nakayama Conjecture in Artin Algebras

• Nakayama Conjecture is a central hypothesis in Artin algebras that connects finite dominant dimension in non-selfinjective algebras with the structure of projective and injective modules.
• Research establishes explicit bounds (e.g., domdim(A) ≤ 2n−2) using tools like syzygy filtration, combinatorial models, and modular techniques to validate the conjecture.
• The conjecture drives advances in serial algebra theory and influences categorical, derived, and geometric approaches, enriching our understanding of representation-finite classifications.

The Nakayama Conjecture is a central hypothesis in the representation theory of Artin algebras, with particular prominence in the paper of serial and cyclic Nakayama algebras. At its core, the conjecture predicts that a non-selfinjective finite-dimensional algebra must have finite dominant dimension, and more generally, it relates the occurrence of vanishing of homological invariants (such as higher Ext groups or the dominant dimension) to the selfinjectivity of the algebra. The conjecture has inspired a substantial body of research tying combinatorial, modular, and categorical structures to invariants such as projective and injective modules, dominant dimension, and representation type.

1. Statement and Historical Context

The classic Nakayama Conjecture asserts that for any non-selfinjective finite-dimensional algebra $A$ (over an algebraically closed field), the dominant dimension of $A$ is finite. If $A$ is selfinjective, all modules are injective, and the conjecture trivially holds. For Artin algebras which are not selfinjective, the conjecture proposes a deep connection between the structure of the module category, the injective resolutions, and the algebra's homological dimensions. This conjecture sits in parallel to related problems, such as the finitistic dimension conjecture, Foulkes' Conjecture in the context of symmetric groups, and the Yamagata conjecture on bounds for dominant dimension (Marczinzik, 2016).

Historically, the conjecture emerged from questions surrounding the homological behavior and classification of Nakayama algebras—those with unique composition series for each indecomposable module—and has been examined in both modular and non-modular settings.

2. Homological Invariants and Characterizations

Key invariants involved in the conjecture are the dominant dimension and projective/injective modules. The dominant dimension is defined by the minimal injective coresolution of the regular module:

$$\operatorname{domdim}(A) = \sup\{n \mid I_0, I_1, \ldots, I_n \text{ are projective}\} + 1.$$

Explicit bounds on $\operatorname{domdim}(A)$ for Nakayama and monomial algebras are established:

• For a Nakayama algebra with $n$ simple modules and $s$ nonisomorphic indecomposable projective–injectives,

$\operatorname{domdim}(A) \leq 2s,$

with the optimal case for monomial/Nakayama algebras,

$\operatorname{domdim}(A) \leq 2n - 2.$

This result answers a question of Abrar and proves Yamagata’s conjecture in this context (Marczinzik, 2016), with examples showing the bound is attained.

The implication is that algebraic structure—like the number and configuration of projective–injective modules—controls homological dimensions; the interplay is especially transparent in Nakayama algebras given via their Kupisch series.

3. Seriality and Representation-Finite Classification

A major advance in serial algebra theory emerges from the notion of $n$-factor serial modules, leading to a refined partition of representation-finite algebras:

• Every finitely generated indecomposable right module is $i$-factor serial for some $1 \leq i \leq n$.
• An artin algebra $\Lambda$ is representation-finite if and only if $\Lambda$ is right $n$-Nakayama for some $n$ (Nasr-Isfahani et al., 2017).
• Classifications are given for hereditary right $n$-Nakayama algebras, tightly tied to Dynkin diagram types.

This approach not only provides a quantifiable gradation measuring "distance" from uniseriality but also facilitates proofs and reformulations of the Brauer–Thrall conjectures within the serial framework.

4. Combinatorial and Block-Theoretical Connections

In modular representation theory, the Nakayama Conjecture is closely associated with border-strip tableaux and the structure of blocks via $n$-core removals. Advanced combinatorial models—such as generalized border-strip tableaux and character deflation maps—enable explicit formulas for nonzero character values and cancellation patterns:

• The deflation map restricts characters of $S_{mn}$ to $S_m \wr S_n$ and averages over the base group, extracting the $S_n$ part. The value on $g \in S_n$ is given by (Evseev et al., 2012):

$$(\mathrm{Defres}_{S_n}\chi)(g) = \frac{1}{|B|} \sum_{k \in B} \chi(k; g) \cdot \widetilde{w}^{\times n}(k; g),$$

where the sum is over the base subgroup $B = S_m \times \cdots \times S_m$.

• The combinatorial formula for deflated characters generalizes the Murnaghan–Nakayama rule:

$$(\mathrm{Defres}_{S_n}\chi^{\lambda/\mu})(g) = \sum_{T} \operatorname{sgn}(T) =: a_{(\lambda/\mu, \gamma)},$$

with the sum over $m$–border–strip tableaux and sign determined by strip heights.

These combinatorial interpretations yield criteria for character vanishing, essential to the spirit of Nakayama’s original block-theoretic conjecture.

5. Derived, Syzygy, and Cluster-Tilting Approaches

Syzygy filtered algebras and their duals (cosyzygy filtered algebras) provide functorial and categorical frameworks for tracing homological dimensions:

• The syzygy filtered algebra $\text{\textcurrency}(A) = \operatorname{End}_A \left( \bigoplus_{S \in S'} P(S) \right)$ encodes filtered module categories (Sen et al., 1 Jun 2024).
• Defect invariants control the transfer of dominant dimension under syzygy filtration, with the precise relation:

$\operatorname{domdim} A = 2 + \operatorname{domdim} \text{\textcurrency}(A)$

for defect-invariant Nakayama algebras.

Cluster-tilting objects, arising via the Auslander–Iyama correspondence, deliver further categorical insight; e.g., the endomorphism algebra of a generator possesses $2$-cluster–tilting structure for certain minimal Auslander–Gorenstein Nakayama algebras.

6. Quasi-Hereditary Orderings and Combinatorial Criteria

A total ordering on simple modules that turns every Weyl module Schurian and ensures projective modules admit a filtration by Weyl modules is termed a quasi-hereditary ordering (q-ordering). For Nakayama algebras, there is an explicit combinatorial criterion for q-orderings:

• For a Nakayama algebra $A = KQ_n/I$ with generators $g_i$, the ordering is quasi-hereditary iff the maximum of the hood $Hod(g_i)$ is not an interior element for any $g_i$ (Zhang et al., 5 May 2024).
• The Green–Schroll set $\mathcal{X}_A$—modular indices not appearing as interior elements—characterizes quasi-heredity: $A$ is quasi-hereditary iff $\mathcal{X}_A \neq \emptyset$.

The $q$-ordering conjecture, stating $q(A) \leq \frac{2}{3} n!$ for Nakayama algebras, is positively resolved; explicit formulas such as $q(A) = (2/\ell_g)\, n!$ for principal ideal case (generator $g$ of length $\ell_g$) are established.

7. Generalizations, Future Directions, and Relationships to Other Conjectures

The conjecture’s connections extend to derived equivalence and singularity theory:

• Derived equivalence classes of Nakayama algebras with almost separate relations admit combinatorial classification via quipu quivers (Fosse, 2023).
• The derived tame classification relates Nakayama algebras to gentle and skewed-gentle algebras, with singularity category precisely determined by the minimal relations (Bekkert et al., 2019).

Elsewhere, continuous analogues of Nakayama representations over $\mathbb{R}$ and $\mathbb{S}^1$ present new topological and dynamical perspectives, opening avenues for homological analysis and possible counterexamples to classical conjectural bounds (Rock et al., 2022).

In algebraic geometry, the Nakayama–Zariski decomposition’s behavior under the Minimal Model Program is linked to termination of flips—a natural conjecture (Conjecture 1.4 in (Lazić et al., 2023)) predicts that the "negative" part of the decomposition must eventually be eliminated by a sequence of MMP flips, paralleling the disappearance of defects in module-theoretic analogues.

Table: Logical Interrelations of Key Results and the Nakayama Conjecture

Aspect	Core Result/Technique	Implications/Context
Dominant dimension bounds	$\operatorname{domdim}(A) \leq 2n-2$	Verifies finiteness for Nakayama, supports Yamagata conjecture (Marczinzik, 2016)
$n$-Nakayama type	Seriality invariant for modules	Partitions representation-finite algebras, connects to Brauer–Thrall (Nasr-Isfahani et al., 2017)
Quasi-hereditary ordering	Hood/Green–Schroll set criterion	Confirms $q$-ordering conjecture (Zhang et al., 5 May 2024), links structure to combinatorics
Derived/cluster perspectives	Syzygy filtration, defect invariance	Controls homological dimensions, supports classical formulation (Sen et al., 1 Jun 2024)
Combinatorial character rules	Border-strip tableaux, deflation map	Generalizes Murnaghan–Nakayama, tests vanishing conditions (Evseev et al., 2012)

Conclusion

The Nakayama Conjecture remains a focal point in the homological and combinatorial paper of Artin algebras. Rigorous upper bounds on dominant dimension, explicit criteria for quasi-hereditary orderings, and combinatorial character formulas have been established for large classes of Nakayama and related algebras. Functorial constructions such as syzygy filtered and derived equivalences, along with categorical and topological models, further reinforce the conjecture’s predictions and provide new tools for its paper. Progress continues in classifying special homological types (higher Auslander–Gorenstein, dominant Auslander–regular) and in relating categorical invariants to algebraic and combinatorial structure, collectively deepening the understanding of when and how the Nakayama Conjecture holds.