Para-Exceptional Sequences

• Para-exceptional sequences are ordered module sequences that include both exceptional modules and non-homogeneous bricks, broadening classical frameworks in representation theory.
• They utilize binary chain systems and closure operators to construct a graded lattice structure isomorphic to the McCammond–Sulway lattice, refining reflection factorizations in affine Coxeter groups.
• This framework unifies approaches to modeling thick subcategories and noncrossing partitions, providing new insights into the combinatorial and geometric properties of infinite-type algebras.

Para-exceptional sequences are a novel class of ordered module sequences introduced in the representation theory of tame hereditary (Euclidean) algebras to model and realize the combinatorial Garside structure known as the McCammond–Sulway lattice—an enlargement and lattice completion of the noncrossing partition poset in affine Coxeter groups. These sequences generalize classical exceptional sequences by broadening the set of allowed entries to include all non-homogeneous bricks, particularly modules corresponding to the non-homogeneous tubes of tame hereditary categories. The theory provides new links between reflection factorizations, poset combinatorics, and the subcategory structure of infinite-type representation categories, offering a lattice-theoretic perspective that covers phenomena not visible in finite Dynkin type.

1. Definition and Motivation

A para-exceptional sequence is an ordered sequence $(X_1, ..., X_r)$ of modules over a connected tame hereditary algebra, where each $X_i$ is either an exceptional module or a brick from a non-homogeneous tube (the so-called $F$-modules), and for $j > i$,

$$\mathrm{Hom}(X_j, X_i) = 0 = \mathrm{Ext}^1(X_j, X_i).$$

In classical settings, only the exceptional objects (those with division algebra endomorphism rings and vanishing self-extensions) are permitted; para-exceptional sequences enlarge this set by admitting these non-homogeneous bricks, which are not themselves exceptional but are maximally rigid (bricks) in their tube categories.

This relaxed framework enables a representation-theoretic realization of larger combinatorial structures such as the McCammond–Sulway lattice, which can emerge when the noncrossing partition poset fails to be a lattice in affine type. In particular, the presence of para-exceptional sequences ensures closure under certain mutation and chain formation operations that are obstructed in the classical setting (Hanson et al., 17 Oct 2025).

2. Structure and Construction of Para-Exceptional Sequences

Fundamental to the construction is the distinction between the two “alphabets”: $E$, the set of exceptional indecomposables, and $F$, the non-homogeneous bricks. Para-exceptional sequences are thus “brick-brick sequences” indexed by $E \cup F$, ordered so that all hom and extension vanishing properties are satisfied in the prescribed direction.

The construction often relies on the machinery of chain systems—a concept wherein maximal chains in a poset are described by finite words in a chosen alphabet, preserving the ordering and labelling of intervals. Specifically, a binary chain system is one in which each maximal chain is encoded by a word in two distinct elements at each step. The paper establishes that the set of complete para-exceptional sequences forms such a binary chain system, enabling precise combinatorial correspondence with reduced reflection factorizations in affine Coxeter groups (Hanson et al., 17 Oct 2025).

In practice, the sequence construction requires one:

• To select modules from $E \cup F$, allowing repetitions only as permissible by the tube category structure.
• To arrange them in an order so that the vanishing hom-orthogonality conditions are met.
• To associate to each module an “edge label” that underlies the chain system encoding.

3. Para-Exceptional Subcategories and Closure Operators

Associated to every para-exceptional sequence is a para-exceptional subcategory, defined as a closure under a certain operator acting on the wide subcategory generated by the sequence. For a traditional exceptional sequence, this is the wide subcategory $\mathcal{W}$ generated by the modules. In the para-exceptional context, closure is extended to include the non-homogeneous bricks, reflecting the extended reach of the category due to tubes.

This closure operator behaves analogously to lattice completion in posets. Para-exceptional subcategories thus embody the “expanded” combinatorics necessary for modeling the lattice properties of the McCammond–Sulway structure, in contrast to the sometimes incomplete intersection patterns among classical wide subcategories in affine type.

4. Combinatorial Lattice Structures and Garside Property

The principal achievement is the isomorphism between the poset of para-exceptional subcategories (ordered by inclusion) and the McCammond–Sulway lattice, where the latter is designed to fill the gaps left in the noncrossing partition poset for infinite (affine) Coxeter groups. The elements of the para-exceptional subcategory poset are labeled by words in $E \cup F$, and the poset is graded by a natural weight function (with exceptional modules assigned weight $1$, bricks assigned corresponding weights respecting tube structure).

The paper rigorously proves that this poset is a graded lattice and moreover satisfies all axioms required of a combinatorial Garside structure. This reveals a deep connection between the algebraic structure of tame hereditary categories and the geometric group-theoretic constructs in Artin groups and their dual presentations (Hanson et al., 17 Oct 2025).

5. Binary Chain Systems and Reflection Factorization

Under the established binary chain system, each para-exceptional sequence is mapped to a reduced word in a generating set for the Coxeter element $c$ of the associated affine group, with the alphabet now extended to include translation generators $F$ alongside the reflections $T$. In finite type, this recovers the classical correspondence between exceptional sequences and minimal reflection factorization in the group; in affine type, the augmentation via $F$ ensures the interval $[1, c]$ in the larger group is a lattice.

As a consequence, for every para-exceptional sequence, one may unambiguously encode its reflection (or translation) data, leading to a bijection between maximal chains in the para-exceptional poset and reduced factorizations of $c$ in the McCammond–Sulway supergroup.

Alphabet	Sequence Type	Chain System
$T$ (reflections)	Exceptional	Noncrossing chains
$T \cup F$	Para-exceptional	McCammond–Sulway

6. Tube Categories, Regular Modules, and Type $\widetilde{A}$

In tube categories (components of the representation theory corresponding to non-homogeneous tubes), para-exceptional modules are precisely those string modules that realize maximal rigidity (bricks in the tube). The poset of wide subcategories generated by these modules inherits a binary chain system structure, and the closure operator is adapted to reflect the combinatorics of thick subcategories.

In type $\widetilde{A}$, many of the para-exceptional modules align with string modules, and the para-exceptional subcategory poset relates directly to recent work on thick, string, or “soft exceptional” subcategories in the derived category. This extension covers settings where the usual noncrossing partition posets are not lattices due to reducibility or other affine complications (Hanson et al., 17 Oct 2025).

7. Implications and Related Constructions

The representation-theoretic proof that the McCammond–Sulway lattice is a combinatorial Garside structure validates (and recovers) prior group-theoretic and combinatorial approaches in the literature. Para-exceptional sequences also provide a perspective to paper broader lattice and poset structures in infinite-type representation theory, especially their connections to stability conditions, wide subcategory closure, and categorification phenomena.

Comparisons are made to semi-stable wide subcategories [Ingalls–Paquette–Thomas] and thick string subcategories, revealing how the para-exceptional theory generalizes and unifies approaches to Artin group dual presentations, poset completions, and representation-theoretic lattices.

Summary

Para-exceptional sequences transcend the boundaries of classical exceptional sequences by admissibility of non-homogeneous bricks, enabling realization of combinatorial lattices beyond finite type. Through chain systems, closure operators on subcategories, and a graded lattice structure, they establish an isomorphism with the McCammond–Sulway lattice and provide a representation-theoretic proof of its Garside properties. This approach significantly advances the understanding of subcategory lattices in the module theory of tame hereditary algebras and the combinatorics of infinite Coxeter group presentations (Hanson et al., 17 Oct 2025).