DG-Enhanced Schur Algebra Overview

• DG-enhanced Schur algebras are differential graded algebra structures that lift classical Schur algebras by incorporating a stratified cell basis with compatible differentials.
• They merge combinatorial cell constructions with DG techniques to enable advanced categorification, representation theory, and explicit control over deformation phenomena.
• Their structure supports derived equivalences and Morita equivalence frameworks, facilitating applications in block theory, modular representations, and quantum invariants.

A DG-Enhanced Schur Algebra is a differential graded (DG) algebra structure placed on a Schur algebra or its generalizations, allowing homological and stratification properties to be studied in the derived category context. This concept merges the cell algebra stratification and combinatorial basis constructions of Schur-type algebras with differential and graded enhancements, enabling representation-theoretic, categorification, and homological applications. The DG enhancement is motivated by several frameworks such as derived categories, deformation theory, and categorified quantum invariants.

1. Cell Algebra Structure and Stratification Principles

The foundational algebraic framework for DG-enhanced Schur algebras is the cell algebra structure as described for generalized Schur algebras (May, 2016). In the cell algebra setting, an $R$-algebra $A$ is equipped with a basis $$\{C_{s,t}^\lambda : \lambda \in \Lambda, s \in L(\lambda), t \in R(\lambda)\}$$ stratified by a partially ordered set $\Lambda$. The cell properties require that for all $a \in A$,

$$a \cdot C_{s,t}^\lambda \equiv \sum_{s'} r_L(a, \lambda, s, s') C_{s',t}^\lambda \mod A^{>\lambda}$$

and

$$C_{s,t}^\lambda \cdot a \equiv \sum_{t'} r_R(a, \lambda, t, t') C_{s,t'}^\lambda \mod A^{>\lambda}$$

where $A^{>\lambda}$ denotes the ideal generated by higher cells under the partial order. These triangularity properties enable direct combinatorial constructions of irreducible representations via radical quotients of cell modules and ultimately provide quasi-heredity criteria (e.g., all cell modules with nonvanishing bracket imply quasi-heredity).

For the DG-enhanced context, the cell basis and its stratified filtration correspond to comparable graded pieces in a DG algebra, and the differential is typically constructed to preserve or interact compatibly with this filtration. Thus, one obtains in the DG setting "cell modules" as standard objects in the derived or triangulated category, whose irreducible quotients remain parametrized by combinatorial data (tableaux, orbits, double cosets) (May, 2016).

2. Models and Constructions of DG-Enhancements

DG-enhancements are realized by constructing DG algebras whose homological properties reflect and refine the classical Schur algebra structure. Explicit models occur, for example, in the context of the DG-enhancement of the derived category of quasi-coherent sheaves $D(QCoh(X))$, where the homotopy category of a suitable model structure (built from diagrams of DG algebras over affine covers) provides the enhancement (Meazzini, 2018). In this model, for a sheaf $F$, a cofibrant replacement $Q$ in the model category Mod$(A_{(\mathfrak{a})})$ yields

$$\operatorname{REnd}(F) \cong \operatorname{Hom}_{A_{(\mathfrak{a})}}(Q, Q)$$

with differential and graded commutator structure, making it a DG-Lie algebra. Maurer–Cartan elements in this DG-Lie algebra control the deformation theory of $F$ and classify extensions and obstructions in terms of cohomology.

While this context does not construct DG-enhanced Schur algebras directly, it shows the methodology: one models the underlying category by explicit DG algebra objects and associated module categories, rendering derived and deformation-theoretic phenomena computable via DG enhancements. The cell basis and combinatorial parametrization of Schur algebras allow for a compatible lift to the DG field (May, 2016, Kleshchev et al., 2018).

3. Generalized Schur Algebras, Symmetricity, and Morita Equivalence

Generalized Schur algebras exhibit rich structural features critical to DG-enhancement. These include:

• Explicit bases and generators constructed from permutations and combinatorial triples (e.g., rescaled $\eta^{\mathbf{b}}_{r,s}$ elements indexed by $S_d$-orbits).
• Symmetricity conditions, where the existence of a perfect symmetrizing form (especially when the form vanishes or pairs perfectly on specified subspaces) ensures the algebra is symmetric. For such algebras $T_a(n,d)$, this is realized via a central form (e.g., $t(\eta^{\mathbf{b}}_{r,s})$ formulas) (Kleshchev et al., 2018).
• Double centralizer properties, which provide Morita equivalences between Schur algebras (or their blocks) and endomorphism rings of suitably chosen idempotent modules; this is essential for both block theory and the passage to derived equivalences. The form $$\lambda: S(n,d) \to \operatorname{End}_{S(n,d)\xi}(S(n,d)\xi)$$ realizes such a double centralizer for compatible idempotents (Kleshchev et al., 2018).

These features are directly leveraged in the construction of DG-enhanced versions—symmetricity assists the extension of the algebra structure to the DG setting, and double centralizer properties allow the transfer of representation-theoretic and homological data under equivalence of categories.

4. Categorification, DG-Modelling, and Representation Theory

The passage to DG-enhanced Schur algebras is tightly linked to modern categorification programs, stratified homological algebra, and the paper of derived categories of modules. The cell algebra’s stratification aligns well with grading and filtration structures in DG algebras. In the DG setting, standard modules correspond to cell modules, and their radical quotients yield the irreducible objects in the derived category (May, 2016).

In categorification contexts (e.g., KLR algebras), the combinatorial basis, multiplication rules, and block structure are mirrored in DG enhancements where differentials reflect relations between composable morphisms and stratifications. Morita equivalence, driven by double centralizer properties, often survives in DG enhancements, yielding derived and DG-equivalences for blocks and local structures (cf. RoCK blocks and extended zigzag Schur algebras (Kleshchev et al., 2022)).

5. Connections to Deformation Theory and Derived Categories

DG-enhanced structures provide strong control over deformation theory and derived extensions. The philosophy articulated in (Meazzini, 2018) is that Maurer–Cartan elements in a DG-Lie algebra constructed from DG endomorphisms of a generator (for Schur-type modules, via parahoric induction or cellular basis modules) yield the deformation functor and obstructions: $$T^1\mathrm{Def}_F \cong H^1(\operatorname{REnd}(F)),\quad \text{obstructions in } H^2(\operatorname{REnd}(F))$$ This principle extends to DG-enhanced Schur algebra modules, where classical extension groups $\operatorname{Ext}^i(M,N)$ appear as cohomology of the relevant DG endomorphism algebra. The explicit DG enhancement model (e.g., per$(\operatorname{dgEnd}(\mathfrak{P}_V))$ for the derived unipotent block of $p$-adic $\mathrm{GL}_2$ (Berry, 26 Nov 2024)) allows triangulated equivalence between the bounded derived category of module representations and perfect complexes over DG Schur algebras.

6. Applications to Block Theory, Stratification, and Quantum Symmetry

DG-enhanced Schur algebras have substantial impact in modular representation theory, categorification, and quantum topology. The stratification afforded by the cell basis framework assists in block decomposition, and explicit construction of weight idempotents enables blockwise paper and derived equivalence transformations (see idempotents $\ell_a$ with $\sum_a \ell_a = 1$ for block decomposition (Du et al., 13 Jun 2024)). For blocks with cyclic defect or with affine cellular structure, DG-enhanced models yield explicit control over projective and tilting module categories (Berry, 26 Nov 2024, Kleshchev et al., 2022).

In quantum and geometric representation theory, DG enhancements facilitate categorified quantum group actions and geometric Langlands correspondences, as in the geometric construction of generalized Schur algebras via Borel-Moore homology and equivariant K-theory (Luo et al., 27 Nov 2024).

7. DG-Enhanced Schur Algebra: Summary and Perspectives

The DG-enhanced Schur algebra is a DG algebra lifting of the classical (possibly generalized) Schur algebra, structured to retain:

• A stratified cell basis compatible with filtration and grading,
• Multiplicative structure and symmetry properties (central forms, perfect pairings),
• Double centralizer and Morita equivalence phenomena,
• Categorification and derived equivalences through both algebraic and geometric models,
• Explicit connection to deformation theory via DG-Lie algebra control,
• Applicability to block theory and quantum representation categories.

The construction of such enhancements exploits explicit cellular, combinatorial, and geometric presentations, and their derived categories model both the homological and representation-theoretic fiber of Schur-type algebraic structures. The essential features are preserved in the DG enhancement, enabling coherent paper and application across a broad range of modern mathematical domains.