Heisenberg Categories and Diagrammatics

Updated 21 August 2025

Heisenberg categories are diagrammatic monoidal categories that categorify integral forms of Heisenberg algebras using generators like cups, caps, and crossings.
They feature explicit combinatorial bases for morphism spaces, enabling clear translations between diagrammatics and algebraic structures.
Their applications extend to categorical actions on module and derived categories, linking representation theory, geometry, and Kac–Moody categorification.

Heisenberg categories are pivotal diagrammatic monoidal categories and 2-categories that categorify integral forms of Heisenberg algebras, their Fock space representations, and related structures in representation theory and geometry. Originating with Khovanov’s graphical calculus, they underpin a broad programme connecting categorification, higher representation theory, and geometry, most notably by providing canonical categorical lifts of Heisenberg actions on various module and derived categories, as well as explicit diagrammatics for functorial correspondences. Their essential features include diagrammatic presentations with cups, caps, crossings, and dots, an explicit combinatorial basis, and a powerful bridge to Kac–Moody categorifications.

1. Core Definitions and Categorification Principles

Heisenberg categories are strict monoidal (and in 2-categorical extensions, additive) categories generated by two fundamental objects (often denoted $P$ and $Q$ or $Q_+$ and $Q_-$ ), together with morphisms rendered as planar diagrams. These generators correspond at the decategorified level to the creation and annihilation operators (or $p_n$ , $q_n$ generators) of an infinite-rank Heisenberg algebra, governed by the canonical commutation relations $[p_m, q_n] = \delta_{mn}$ .

A key structural aspect is the explicit use of a graphical calculus with local relations, where:

Upward and downward oriented strands correspond to $P$ and $Q$ .
Cups and caps encode (bi)adjunctions.
Crossings are either symmetric group or Hecke algebra generators—permitting $q$ -deformation.
Dots and tokens (possibly labeled by a Frobenius algebra or group action) encode polynomial actions and further algebraic data.

There are multiple variants, including degenerate, $q$ -deformed, and Frobenius–enriched categories, all unified by the common formalism of diagrammatic presentations with explicit relations (e.g., nilHecke/Hecke quadratic and braid relations, dot-sliding, and pivotality) (Wang et al., 2013, Licata et al., 2011, Savage, 2018, Brundan et al., 2018, Brundan et al., 2020).

The Grothendieck group of the Karoubi envelope (idempotent completion) of a Heisenberg category is canonically isomorphic to an integral form of the Heisenberg algebra. Canonical isomorphisms such as

$T(p^{(n)}_i) = [P_i^{(n)}]\,,\quad T(q^{(n)}_i) = [Q_i^{(n)}]$

realize divided powers and combinatorial bases, providing faithful integral lifts of the algebraic structures (Cautis et al., 2010, Wang et al., 2013, Brundan et al., 2018).

2. Diagrammatics, Morphism Bases, and Presentations

Diagrammatic presentations are foundational. Objects are sequences of $P$ and $Q$ , and morphisms are generated by planar diagrams modulo local relations:

Generator	Diagrammatic Representation	Algebraic Role
Upward/Downward	Vertical arrow ↑ or ↓	$P$ / $Q$ functor/application
Crossing	Strand intersection	Symmetric/Hecke generator, commutation
Dot/Token	Dots/labels on strands	Polynomial actions, algebra elements (e.g., $x$ )
Cup/Cap	Semi-circular arcs	(Bi)adjunction, pivotal duality

Local relations include:

Commutation and braid relations for crossings,
Dot-sliding and token relations with explicit dependence on possible Frobenius or group structures,
Bubble or curl relations evaluating closed loops to central elements or scalars,
Infinite Grassmannian relations (expressing determinantal identities akin to cohomology of Grassmannians).

A haLLMark of these presentations is the existence of an explicit combinatorial basis for morphism spaces: any morphism can be reduced—via diagrammatic moves—to a unique normal form, producing a free module over symmetric functions $\operatorname{Sym} \otimes \operatorname{Sym}$ with basis indexed by “reduced matchings” and “dot placements” (Brundan et al., 2018, Brundan et al., 2018, Brundan et al., 2020).

In degenerate and quantum variants, further parameters such as central charge $k \in \mathbb{Z}$ and quantum deformation $q$ (with $z=q-q^{-1}$ , $t=-z^{-1}$ ) parameterize central extensions and deformations of the algebraic structures (Brundan et al., 2018).

In Frobenius–enriched and higher-level settings, tokens on strands record the action of elements in a symmetric Frobenius (super)algebra $A$ , and additional "dual dot" operators and higher dot-circle relations encode level data and extend the diagrammatics to a lattice Heisenberg algebra (Savage, 2018, Brundan et al., 2020, Brundan et al., 2020).

3. Representation-Theoretic and Geometric Realizations

Heisenberg categories act via categorical (bi)adjoint functors on a broad class of module categories and derived categories. Notable realizations include:

Actions on the categories of modules over symmetric groups and Hecke algebras using induction and restriction functors; in this context, the upward and downward generators correspond respectively to induction and restriction, and the diagrammatic calculus captures the natural transformations between their compositions (Licata et al., 2011, Queffelec et al., 2017, Brundan et al., 2018).
Constructions of categorical Heisenberg actions on the derived categories of coherent sheaves on Hilbert schemes of points on minimal resolutions of quotient singularities ( $\operatorname{Hilb}^n(X)$ for $X=\widehat{\mathbb{C}^2/G}$ ), realized via Fourier–Mukai kernels corresponding to "add a point" or "remove a point" along an exceptional cycle. This "lifts" the classical Fock space representation (observed in cohomology or $K$ -theory) to a categorical level (Cautis et al., 2010, Licata et al., 2011).
Categorification of Heisenberg actions on the Grothendieck groups of category $\mathcal{O}$ for cyclotomic rational double affine Hecke algebras (DAHAs), rational Cherednik algebras, and their cyclotomic variants, where the functors constructed (often biadjoint) correspond to adding/removing boxes (or particles), and respect geometric and representation-theoretic filtrations (Shan et al., 2010, Bezrukavnikov et al., 5 Aug 2024).
Graphical calculi for induction and restriction with explicit correspondence to the actions of degenerate (cyclotomic) Hecke algebras, general linear groups, and their (parabolic) induction/restriction functors, often employing explicit idempotents and token-labeled diagrams (Mackaay et al., 2017, Licata et al., 2011).

4. Bridging Kac–Moody Actions, Cyclotomic Quotients, and Canonical Bases

Heisenberg categorifications serve as an interface to Kac–Moody 2-categories and quantum group categorification. Via spectral decompositions and eigenfunctors (for dot actions), every abelian module over a Heisenberg category which satisfies suitable finiteness and nilpotency conditions can be equipped with a 2-representation structure for an associated Kac–Moody 2-category (e.g., of affine type $A_\infty$ or $A_{p-1}^{(1)}$ ) (Brundan et al., 2019, Queffelec et al., 2017, Brundan et al., 20 Aug 2025).

Explicit diagrammatic and functorial correspondences between induction/restriction and Kac–Moody functors (such as the $i$ -induction functor for content $i$ in symmetric group blocks) are available. Abstract properties of the categorical action (pivotality, biadjunctions, explicit bubble generating functions) are crucial in producing isomorphisms and derived equivalences between weight subcategories (e.g., blocks of symmetric groups), as predicted by the combinatorics of affine crystals and the structure of the Heisenberg Fock space (Brundan et al., 20 Aug 2025).

Cyclotomic quotients of Heisenberg categories—obtained by imposing polynomial relations on dot operators or bubbles—produce module categories whose Grothendieck rings categorify (truncated or minimal) highest/lowest-weight representations of Kac–Moody or Heisenberg algebras, giving a precise categorical realization of Fock or minimal modules. These quotients link to cyclotomic Hecke and Cherednik algebras and play a central role in the construction of categorical representations, vertex operator actions, and projective functors (Brundan, 2017, Brundan et al., 2019, Mackaay et al., 2017, Brundan et al., 2018).

The canonical basis afforded by indecomposable 1-morphisms in the Heisenberg 2-category provides an explicit lift of the Kazhdan–Lusztig or PBW bases, yielding strong integral and combinatorial control on decategorification (Cautis et al., 2010, Wang et al., 2013).

5. Extensions: Frobenius Structures, Generalizations, and Basis Theorems

Recent generalizations enrich Heisenberg categories with symmetric Frobenius superalgebras (Frobenius–Heisenberg categories), broadening the range of categorified Heisenberg/lattice-type algebras, and allowing the construction of categories $\mathcal{H}\mathrm{eis}_{F,k}$ , $\mathcal{H}\mathrm{eis}_k(A;z,t)$ with token-decorated strands, inversion relations reflecting the defect in up-down crossings, and corresponding basis theorems (Savage, 2018, Brundan et al., 2020, Brundan et al., 2020).

A central technical achievement is the derivation of explicit combinatorial bases for morphism spaces in all main Heisenberg category variants. These bases consist of diagrams (reduced matchings with tokens and dots) that, after imposing all local relations, enumerate a free spanning set over centers (such as symmetric functions or cohomology rings). For each pair of objects $X, Y$ , $\operatorname{Hom}(X, Y)$ is a free module over $\operatorname{Sym} \otimes \operatorname{Sym}$ (or similar), and the endomorphism algebra of the unit object is canonically isomorphic to this central algebra (Brundan et al., 2018, Brundan et al., 2018, Brundan et al., 2020).

The basis theorems, often established via inductive “sideways crossing” arguments and reduction to cyclotomic quotients, guarantee both the freeness and linear independence of such diagrammatic bases. This is crucial for explicit computations, verification of categorification claims, and connections to invariant theory and homology (Brundan et al., 2018, Brundan et al., 2020).

6. Applications, Consequences, and Broader Impact

Heisenberg categories and their categorical actions have synthesized various strands in modern representation theory, geometry, and low-dimensional topology:

They form the categorical underpinnings of the geometric actions on Hilbert schemes, giving rise to Fock space modules, and are essential to the paper of vertex algebras and geometric representation theory (e.g., refinement of Nakajima’s constructions) (Cautis et al., 2010, Licata et al., 2011).
In type $A$ modular representation theory, abstract properties of Heisenberg and Kac–Moody categorifications explain and “prove” major conjectures, such as Broué’s Abelian Defect Conjecture for symmetric groups, by establishing derived equivalences of blocks via categorical Morita theory facilitated by explicit diagrammatics and functorial correspondences (Brundan et al., 20 Aug 2025).
The diagrammatic frameworks unify and systematize the construction of canonical bases, branching rules, and crystal combinatorics, and give diagrammatic proofs of classical and novel representation-theoretic identities (Queffelec et al., 2017, Licata et al., 2011).
The theory’s flexibility allows for efficient and general presentations using inversion relations, minimal generator/relations lists, and connections to affine oriented Brauer categories, HOMFLY–PT skein categories, and more (Brundan, 2017, Brundan et al., 2018).
The algebraic and geometric realization of the categorical actions leads to categorical generalizations of the boson–fermion correspondence, strong links to knot homologies, and the categorical extension of the Stone–von Neumann theory for Heisenberg groups (Gomez et al., 2022).

Heisenberg categories, with their explicit basis theorems, combinatorial presentability, and broad range of module/derived categorical actions, have established themselves as central structures in higher representation theory and categorification, with ongoing extensions to new algebraic, geometric, and topological settings.