Left Schur Subcategories in Rep Theory

• Left Schur subcategories are extension-closed subcategories of a length abelian category where each simple object is left Schurian, meaning nonzero morphisms from simples are injective.
• They provide a unifying framework that links monobricks with torsion-free classes and wide subcategories, enabling new classification approaches in categorical representation theory.
• Their applications span module categories, strict polynomial superfunctors, and generalized Schur algebras, impacting modular representation and higher representation theory.

A left Schur subcategory is an extension-closed subcategory of a length abelian category in which every simple object is “left Schurian,” meaning any morphism from a simple object to any object in the subcategory is either zero or injective. This notion unifies and generalizes the theories of torsion-free classes and wide subcategories, providing a categorical and combinatorial framework that links the structure of bricks (simple objects with division endomorphism ring) to the larger architecture of abelian categories and their substructures. The concept arises in various contexts: module categories, strict polynomial (super)functor categories, generalized Schur algebras, and recollements. Left Schur subcategories are instrumental in modular representation theory, categorical structures on functors, higher representation theory, and the paper of cluster categories.

1. Definition and Characterization via Monobricks

A left Schur subcategory $\mathcal{E}$ of a length abelian category $\mathcal{A}$ is defined by two properties:

• $\mathcal{E}$ is extension-closed.
• Every simple object in $\mathcal{E}$ is left Schurian: for any simple $S \in \mathcal{E}$ and $X \in \mathcal{E}$, any nonzero morphism $S \to X$ is injective.

The fundamental result is that left Schur subcategories correspond bijectively to monobricks. A monobrick is a set $\mathcal{M}$ of bricks in $\mathcal{A}$ such that for any $X, Y \in \mathcal{M}$ and any nonzero morphism $f: X \to Y$, the map $f$ is a monomorphism. The bijection is realized as:

• $\mathcal{E} \mapsto \mathrm{sim}\,\mathcal{E}$, the set of all simple objects in $\mathcal{E}$, which forms a monobrick.
• $\mathcal{M} \mapsto \mathrm{Filt}(\mathcal{M})$, the smallest extension-closed subcategory containing $\mathcal{M}$, which is a left Schur subcategory (Enomoto, 2020).

Torsion-free classes and wide subcategories are special types of left Schur subcategories. If $\mathcal{M}$ is cofinally closed (that is, for every brick $N \notin \mathcal{M}$ admitting an injection into some $M \in \mathcal{M}$, there is a non-injective morphism from $N$ to another $M' \in \mathcal{M}$), then $\mathrm{Filt}(\mathcal{M})$ is a torsion-free class. When $\mathcal{M}$ is a semibrick (no nonzero morphisms between different elements), $\mathrm{Filt}(\mathcal{M})$ is wide.

In the Nakayama algebra case, all left Schur subcategories coincide with subcategories closed under extensions, kernels, and images, and their enumeration is given by the large Schröder number (Enomoto, 2020).

2. Construction and Gluing in Recollement Contexts

The construction of left Schur subcategories admits a robust gluing formalism in recollements of length abelian categories (Zhang et al., 18 Oct 2025). Let

$\xymatrix@!C=2pc{ \mathcal{Y} \ar@{>->}[rr]^{i_*} && \mathcal{X} \ar@<-4mm>@{->>}[ll]_{\,i^*} \ar@{->>}[rr]^{j^*} && \mathcal{Z} \ar@{>->}[ll]^{j_!}\ar@{>->}[ll]_{j_*} }$

be a recollement. Suppose $\mathcal{E}_{\mathcal{Y}} \subset \mathcal{Y}$ and $\mathcal{E}_{\mathcal{Z}} \subset \mathcal{Z}$ are left Schur subcategories. Provided that $i^!$ is exact, the glued subcategory

$$\mathcal{E}_{\mathcal{X}} = \left\{ X \in \mathcal{X}\;\middle|\; j^*X \in \mathcal{E}_{\mathcal{Z}},\; i^!X \in \mathcal{E}_{\mathcal{Y}} \right\}$$

is left Schur if and only if both $$\mathcal{E}_{\mathcal{Y}}, \mathcal{E}_{\mathcal{Z}}$$ are left Schur. This follows from the fact that monobricks glue accordingly: $$\mathcal{M}_{\mathcal{X}} = i_*(\mathcal{M}_{\mathcal{Y}}) \sqcup j_*(\mathcal{M}_{\mathcal{Z}})$$ and $$\mathcal{E}_{\mathcal{X}} = \operatorname{Filt}(\mathcal{M}_{\mathcal{X}})$$. This technique extends to wide subcategories and torsion-free classes under similar or weaker exactness assumptions.

An explicit construction is given for cofinally closed monobricks in a recollement: starting from cofinally closed monobricks in $\mathcal{Y}$ and $\mathcal{Z}$, their glued union yields a cofinally closed monobrick in $\mathcal{X}$, and hence a torsion-free class (Zhang et al., 18 Oct 2025).

3. Left Schur Subcategories in Advanced Functor Categories and Generalizations

In the context of strict polynomial superfunctors, the indecomposable Schur superfunctors (as images of natural transformations involving partitions and Hopf superalgebra structure) generate a natural left Schur subcategory (Axtell, 2014). This subcategory serves as the categorical span of the indecomposables, and more complicated bisuperfunctors admit canonical filtrations where subquotients decompose as tensor products of Schur superfunctors, emphasizing the centrality of the left Schur subcategory.

Similarly, in generalized Schur algebras, left Schur subcategories are modules obtained via left generalized Schur algebra $B_L$ or, more precisely, modules filtered via idempotents associated to partitions. The theory of these subcategories, their structure, and the classification of irreducibles depends highly on the base field's characteristic. The filtration stratifies irreducible representations, with the left Schur subcategory picking out those driven by the left multiplication structure and idempotent layers (May, 2016).

In the representation theory of rigid modules and Schur roots over symmetrizable Cartan matrices, the parametrization of left finite bricks by real Schur roots aligns with the understanding of left Schur subcategories as extension-closed subcategories built from bricks with division endomorphism rings and rigidity properties (Geiß et al., 2018).

4. Relations to Torsion-Free Classes, Wide Subcategories, and $\tau$-Tilting Reduction

Left Schur subcategories simultaneously generalize torsion-free classes and wide subcategories. The monobrick classification unifies both: cofinally closed monobricks correspond to torsion-free classes, and semibricks correspond to wide subcategories. These bijections permit finiteness results formerly proven via $\tau$-tilting theory to be deduced directly from the structure of bricks (Enomoto, 2020).

In the language of $\tau$-tilting reduction, left finite wide subcategories—also called left Schur subcategories in related literature—appear as the Serre subcategories of these wider abelian subcategories, especially in the context of $\tau$-perpendicular subcategories and the categorification program for $\tau$-cluster morphism categories. Chains of reductions of this type remain in the class of left Schur (i.e., Serre) subcategories (Buan et al., 2021).

5. Connections to Cell and Representation Theory

In the context of cellular and asymptotic Schur algebras, left Schur subcategories manifest via the classification of left cell representations, block decompositions, and the canonical cellular basis (following Graham-Lehrer). The existence of left cell representatives in the module category, their orthogonality, and uniqueness mirror the structural role of left Schur subcategories within these algebraic environments (Cui et al., 2023). These ideas tie to classical Kazhdan-Lusztig theory, with the subcategories structured by left cells encapsulating indecomposable "building blocks" for the larger module category, now extended to $q$-Schur and asymptotic Schur algebras.

In the theory of 2-categories arising from projective functors for star algebras, the restriction to left cell 2-subcategories and their simple transitive 2-representation theory reflects a "Schur-type" mechanism: the subcategory is generated from the cell data, with classification results indicating that—for low rank cases—all simple transitive 2-representations are cell 2-representations, and in higher rank, more intricate Schur-type subcategories arise, conjecturally in bijection with set partitions (Zimmermann, 2018).

6. Yoneda-Type Constructions and Further Generalizations

A categorical approach is available via restricted Yoneda embeddings: given an algebra $A$ and a subcategory $\mathcal{B}$ of cyclic modules, the restricted Yoneda functor embeds ${}_A\mathsf{Mod}$ into a functor category $[\mathcal{B}^{op}, \mathrm{Vec}_k]$, with an explicit left adjoint. The essential image consists exactly of those modules with specified Schur-like properties—e.g., being generated by their $\mathcal{B}$-invariants—and this construction often recovers known categories such as weight modules, Harish-Chandra modules, and modules for Mickelsson step algebras (Fillmore et al., 17 Jul 2025). In specializations where the objects of $\mathcal{B}$ have division endomorphism rings, the image corresponds to left Schur subcategories in the classical sense.

7. Mathematical Formulations and Methods

The key technical ingredients span several contexts:

• For monobricks to left Schur subcategories: $\mathcal{E} \mapsto \mathrm{sim}\,\mathcal{E}$ and $\mathcal{M} \mapsto \mathrm{Filt}(\mathcal{M})$ (Enomoto, 2020).
• In recollement: the glued subcategory $$\mathcal{E}_{\mathcal{X}} = \{X \mid j^*X \in \mathcal{E}_{\mathcal{Z}},\; i^!X \in \mathcal{E}_{\mathcal{Y}}\}$$, with explicit monobrick union and extension-gluing properties (Zhang et al., 18 Oct 2025).
• In strict polynomial (super)functor categories: indecomposability of Schur superfunctors and explicit module construction via Hopf superalgebra operations (Axtell, 2014).
• In cellular algebra contexts: left cell representations, canonical bases, and computations of multiplicities and block decompositions all provide categorical subdivisions that align with Schur-theoretic notions (Cui et al., 2023).

These form the algebraic and combinatorial substrate for the modern theory of left Schur subcategories, enabling a unified treatment of important classes of subcategories in module and functor categories as well as highly structured representation–theoretic environments.