Cyclotomic Quotient Categories

• Cyclotomic quotient categories are monoidal or 2-categories derived from affine/free categories by enforcing cyclotomic polynomial relations, categorifying rings like ℤ[x]/(Φₙ(x)).
• They provide diagrammatic and homologically structured models that underpin modern higher representation theory and enable categorical actions of quantum groups.
• Key examples include cyclotomic KLR algebras, Brauer categories, and Heisenberg 2-categories, all characterized by explicit bases and functorial equivalences.

A cyclotomic quotient category is a monoidal or 2-category constructed as a quotient of an "affine" or "free" monoidal category by a tensor ideal generated by cyclotomic-type relations, typically involving the vanishing of a cyclotomic polynomial in a key dot or operator generator. Cyclotomic quotient categories serve as diagrammatic, homologically well-structured categorifications of commutative or noncommutative rings defined via cyclotomic relations—most notably, cyclotomic integer rings, cyclotomic Hecke algebras, and their categorified quantum and classical generalizations. These constructions play a central role in modern higher representation theory, categorical actions of quantum groups, and the categorification program, connecting algebraic, diagrammatic, and geometric representation theory.

1. Foundational Construction of Cyclotomic Quotient Categories

Cyclotomic quotient categories arise by imposing cyclotomic (or more general polynomial) relations in a monoidal or 2-category that is freely generated by algebraic or diagrammatic data. The core procedure is as follows:

• Start with an affine/free category: Examples include graded module categories over Nichols or nil-Hecke algebras, diagrammatic web categories with dot/affine generators, or Kac-Moody categorifications such as KLR (quiver Hecke) or Heisenberg categories.
• Define a dot or loop generator (e.g., $x$ in a web or Brauer category, $X$ in an affine Hecke-type setup, or $d_k$ in a braided Hopf algebra): These elements encode the "affine" structure.
• Impose a cyclotomic relation: Form a (right, left, or two-sided) tensor ideal generated by the vanishing of a monic polynomial in this generator, for example,

$f(x) = (x - u_1)\cdots(x - u_\ell) = 0,$

or, more generally, the $n$th cyclotomic polynomial

$$\Phi_n(x) = \prod_{k \in (\mathbb Z/n\mathbb Z)^\times} (x - \zeta^k).$$

• Form the quotient: The category is then the quotient of the original by this ideal. It inherits the monoidal (and often triangulated or 2-categorical) structure of the parent.

This method appears in many guises, including the cyclotomic quotient of:

• The stable category of graded modules over a braided Hopf algebra for categorifying $\mathbb{Z}[x]/(\Phi_n(x))$ (Laugwitz et al., 2018).
• The affine web, Kauffman, Brauer, oriented Brauer categories, with the ideal generated by $f(X)$ or $f(x)$ for varying $f$ (Song et al., 19 Jun 2024, Gao et al., 2020, Rui et al., 2023, Brundan et al., 2014).
• Quiver Hecke (KLR) superalgebras to produce cyclotomic KLR algebras and their projective module categories (Kang et al., 2012).
• Heisenberg or Kac-Moody 2-categories via the imposition of cyclotomic bubble and dot conditions (Brundan et al., 2019).

A unifying property across examples is that such cyclotomic quotient categories categorify modules or algebras defined by cyclotomic-type relations, providing diagrammatic and higher categorical structure with canonical basis and duality features.

2. Archetypal Examples in Diagrammatic and Algebraic Settings

Cyclotomic Monoidal Categories: $\mathcal{O}_n$ and N-Complexes

For $n \geq 2$, consider the category of $\mathbb{Z}$-graded modules over the algebra

$$H_n = \frac{\Bbbk\langle d_1,\dots,d_t \rangle}{(d_1^{p_1},\dots, d_t^{p_t}, [d_i, d_j])}$$

where the $d_i$ are homogeneous, and $p_i$ are the distinct prime factors of $n$. The stable category $\underline{H}_n\mathrm{gmod}$, after quotienting by the ideal generated by induced modules $V_k$ associated to the cyclotomic decomposition, becomes a triangulated monoidal category $\mathcal{O}_n$. Its split Grothendieck ring is canonically isomorphic to the cyclotomic integer ring $\mathbb{Z}[x]/(\Phi_n(x))$ (Laugwitz et al., 2018, Mirmohades, 2018). Analogous constructions hold for $N$-complexes, where the cyclotomic quotient ensures that Kapranov's $q$-commutativity property yields a closed monoidal category precisely over $\mathbb{Z}[q]/(\Phi_n(q))$ (Mirmohades, 2018).

Cyclotomic Brauer/Kauffman/Oriented Brauer/Schur/Web Categories

Numerous diagrammatic monoidal categories (e.g., affine Brauer, Kauffman, oriented Brauer, web, $q$-Schur categories) are defined by local diagrammatic generators, including a dot operator. The cyclotomic quotient is given by imposing $f(x)=0$ for a specified polynomial $f$ (Gao et al., 2020, Rui et al., 2023, Brundan et al., 2014, Song et al., 19 Jun 2024, Shen et al., 14 Apr 2025). In all such cases, the cyclotomic category inherits a diagrammatic basis "as large as possible", with the rank predicted by the number of allowed dot patterns after enforcing the cyclotomic relation.

Cyclotomic Quiver Hecke (KLR) Superalgebras and Quantum Groups

Quiver Hecke algebras (KLR) and their super versions are $\mathbb{Z}$- or $\mathbb{Z}_2$-graded algebras whose projective module categories admit a categorical $\mathfrak{g}$-action. Cyclotomic quotients $R^\Lambda(\beta)$ are obtained by imposing $a_i^\Lambda(x_1) = 0$, where $a_i^\Lambda$ is a cyclotomic polynomial depending on a dominant weight $\Lambda$. The resulting categories of projective modules categorify integrable highest-weight $U_q(\mathfrak{g})$-modules (Kang et al., 2012).

Cyclotomic Heisenberg and Kac-Moody 2-Categories

Heisenberg and Kac-Moody 2-categories admit cyclotomic quotients arising from boundary dot and bubble annihilation relations. The resulting generalized cyclotomic quotients can be canonically identified via diagrammatic monoidal functors, yielding equivalences of certain module categories and underpinning the Brundan-Kleshchev isomorphism between cyclotomic Hecke and KLR algebras (Brundan et al., 2019).

3. Basis Theorems and Admissibility Conditions

In each family of cyclotomic quotient categories, there is a fundamental basis theorem: the morphism spaces admit explicit, diagrammatic, integral bases parameterized by combinatorial data (e.g., dot patterns, partitions, tableaux), subject to the cyclotomic relations cutting off the allowable dot length (at most degree $\ell-1$ for relation of degree $\ell$).

• Affine Kauffman and Brauer categories: Basis consists of normally ordered, reduced, totally descending dotted diagrams, with at most $a-1$ dots per strand for cyclotomic degree $a$ (Gao et al., 2020, Rui et al., 2023).
• Web and $q$-Schur categories: Integral bases given by "elementary ribbon" or "chicken-foot" diagrams, parametrized by matrices of partitions with partition lengths controlled by $\ell$ (Song et al., 19 Jun 2024, Shen et al., 14 Apr 2025).
• Oriented Brauer: Basis formed by normally ordered dotted diagrams with at most $\ell-1$ dots per strand after imposing $f(x)=0$ (Brundan et al., 2014).

The admissibility of the cyclotomic parameters (e.g., $u$-admissibility for families of relation scalars) determines when the expected basis remains free and of maximal rank; failure leads to unwanted linear dependencies (Gao et al., 2020, Rui et al., 2023).

4. Grothendieck Rings and Categorification of Cyclotomic Algebras

Cyclotomic quotient categories provide categorical models for cyclotomic rings and algebras:

• Grothendieck ring computation: For categories such as $\mathcal{O}_n$, $K_0(\mathcal{O}_n)$ is isomorphic to the target cyclotomic ring (e.g., $\mathbb{Z}[x]/(\Phi_n(x))$) (Laugwitz et al., 2018, Mirmohades, 2018). The grading-shift functor corresponds to multiplication by the base variable (e.g., $q$ or $x$).
• Algebraic realization: For Brauer-type and $q$-Schur-type cyclotomic quotients, endomorphism algebras of specific objects are isomorphic to cyclotomic (BMW, Nazarov–Wenzl, Hecke, $q$-Schur) algebras (Gao et al., 2020, Rui et al., 2023, Shen et al., 14 Apr 2025).
• Categorification: Cyclotomic KLR quotient categories categorify highest weight $U_q(\mathfrak{g})$-modules and carry canonical bases matching global crystal bases (Kang et al., 2012).

These identifications are functorial under suitable monoidal functors into module or bimodule categories, and the appearance of canonical bases is a universal feature (e.g., in the case of type $D$ Soergel bimodules and nil-Brauer cyclotomic quotients (Bodish et al., 19 Oct 2024)).

5. Structural and Representation-Theoretic Consequences

Cyclotomic quotient categories underpin essential phenomena across categorical representation theory:

• Cellular and quasi-hereditary structures: The explicit bases are compatible with cellular algebra structures, enabling computation of decomposition matrices, paper of tilting and standard objects, and link to Kazhdan–Lusztig theory (Rui et al., 2023, Shen et al., 14 Apr 2025).
• 2-Representations and Schur-Weyl duality: Cyclotomic quotients realize higher categorical Schur–Weyl dualities, connecting diagrammatic (e.g., web or BMW) categorifications to module categories over quantum or classical Lie algebras via explicit functors (Rui et al., 2023, Shen et al., 14 Apr 2025, Bodish et al., 19 Oct 2024).
• Morita and equivalence results: The algebra of endomorphisms in cyclotomic web categories is often Morita equivalent to (or realizes an idempotent truncation of) a higher Schur algebra or $W$-algebra (Song et al., 19 Jun 2024).
• Universal 2-categorical properties: Cyclotomic quotients realize universal actions—any module category with suitable finiteness admits an action of the corresponding Kac–Moody 2-category, with the cyclotomic quotient providing the universal model (Brundan et al., 2019).

6. Examples and Applications

Notable concrete examples include:

• For $n=2$, the cyclotomic quotient of the stable module category over $H_2$ categorifies the integers via $\mathbb{Z}[q]/(q+1)$, with explicit indecomposables and triangles (Laugwitz et al., 2018).
• The cyclotomic quotient of the degenerate Heisenberg category yields, under suitable identification, the modular representation category of the symmetric group (Brundan et al., 2019).
• In cyclotomic web categories, endomorphism algebras realize Dipper–James–Mathas cyclotomic $q$-Schur algebras, but now with a fully diagrammatic basis and presentation (Shen et al., 14 Apr 2025).
• For cyclotomic nil–Brauer categories, the Grothendieck ring matches a restricted $\imath$-quantum group module with explicit canonical basis elements realized by diagrammatic idempotents (Bodish et al., 19 Oct 2024).
• Cyclotomic KLR superalgebra quotients provide highest-weight categorification for quantum Kac–Moody modules, with strong perfect bases and explicit functors matching categorical and algebraic Chevalley generators (Kang et al., 2012).

7. Broader Connections, Structural Properties, and Future Directions

Cyclotomic quotient categories provide a universal unifying framework for categorical structures controlled by cyclotomic or root-of-unity phenomena:

• Their appearance in the categorification of quantum groups, $W$-algebras, and tensor categories—via diagrammatic, module, or bimodule presentations—demonstrates their central role in higher representation theory (Gao et al., 2020, Song et al., 19 Jun 2024, Shen et al., 14 Apr 2025).
• The theory extends to variant types (orthogonal, symplectic, quantum symmetric pairs) and positive characteristic settings.
• The universal property elucidated by the equivalence of cyclotomic Kac–Moody 2-category and cyclotomic Heisenberg category representations significantly clarifies the connection among diagrammatic, algebraic, and categorical approaches in modern categorical representation theory (Brundan et al., 2019).
• Ongoing directions include cyclotomic Schur categories, affine and nilpotent variants, and explicit connections to canonical and tilting module theory in type $A$, $B$, $C$, and $D$ settings.

A plausible implication is that cyclotomic quotient categories continue to serve as fundamental categorical objects not just as algebraic gadgets but as cornerstones for the geometric and homological underpinnings of categorical representation theory, with further applications to the theory of canonical bases, tensor product categorification, and modular representation theory.