Categorical representation theory

Lens

MotivationH

Why replace numbers by objects?

Four stories, from four corners of mathematics, share one move.

Combinatorics. To prove $\binom{n}{k} = \binom{n}{n-k}$ you can compute, or you can exhibit a bijection between $k$ -subsets and their complements. The bijection says more: it is a reason, and it can be compatible with symmetries that the numbers do not see.
Topology. Betti numbers count holes. Around 1925, Emmy Noether, with Heinz Hopf and Leopold Vietoris, emphasised that they are the ranks of homology groups — and groups, unlike numbers, have maps between them, so a continuous map induces a homomorphism. The Euler characteristic $\chi = \sum (-1)^i b_i$ became the shadow of a richer object.
Representation theory. The Kazhdan–Lusztig polynomials (1979) were conjectured to have non-negative coefficients. Elias and Williamson (2014) proved it for every Coxeter group by showing that the coefficients are dimensions of spaces of maps between Soergel bimodules — and dimensions cannot be negative.
Knot theory. The Jones polynomial (1984) is a powerful knot invariant, but whether it detects the unknot is still unknown. Khovanov (1999–2000) found bigraded homology groups whose graded Euler characteristic is the Jones polynomial; Kronheimer and Mrowka (2011) proved that Khovanov homology does detect the unknot.

In each story a number (or polynomial) is recognised as the shadow — the decategorification — of an object (a set, a group, a vector space, a complex), and an equation between numbers as the shadow of an isomorphism. Going back up is categorification. Categorical representation theory applies this to representation theory itself: instead of a group or algebra acting on a vector space, it studies actions on categories, and representation-theoretic structures on categories such as tensor and fusion categories.

IntuitionH

A ladder, and its shadow map

Picture a ladder:

\text{numbers} \;\rightsquigarrow\; \text{sets, vector spaces} \;\rightsquigarrow\; \text{categories} \;\rightsquigarrow\; \text{2-categories} \;\rightsquigarrow\; \cdots

Going down a rung forgets: a finite set to its cardinality, a vector space to its dimension, a category to its Grothendieck group $K_0$ , the group of isomorphism classes with sums. Going up is not a functor or an algorithm; it is a search, and a structure may have many categorifications or none known.

A representation of a group $G$ on a vector space $V$ is a homomorphism $G \to GL(V)$ . One rung up, a categorical action of $G$ on a category $\mathcal{C}$ is a family of autoequivalences $F_g : \mathcal{C} \to \mathcal{C}$ with coherent isomorphisms $F_g F_h \cong F_{gh}$ ; passing to $K_0(\mathcal{C})$ gives an ordinary representation. Likewise the representation ring of $G$ is the shadow of the representation category $\operatorname{Rep}(G)$ , with $\otimes$ as the categorified multiplication.

Analogy, not a formal construction

Decategorification is like reading a musical score by counting how many notes are written on each page. The counts are honest information — they add up correctly when pages are combined — but two very different pieces can have the same counts, and you cannot hear the music from them. The analogy is informal: the precise statements are about Grothendieck groups below.

DefinitionsH

Definitions

Throughout, $k$ is a field, algebraically closed of characteristic $0$ when we speak of fusion categories.

Definition— Grothendieck group

The Grothendieck group $K_0(\mathcal{A})$ of an essentially small abelian category $\mathcal{A}$ is the abelian group generated by symbols $[X]$ , one for each isomorphism class of objects, with a relation $[B] = [A] + [C]$ for every short exact sequence $0 \to A \to B \to C \to 0$ . For an additive category one uses only split sequences ( $[A \oplus C] = [A] + [C]$ ); for a triangulated category, one relation per distinguished triangle. If $\mathcal{A}$ has a biexact tensor product, $[X][Y] = [X \otimes Y]$ makes $K_0(\mathcal{A})$ a ring.

Definition— tensor category

A tensor category over $k$ (in the sense of Etingof–Gelaki–Nikshych–Ostrik) is a locally finite $k$ -linear abelian category with a bilinear monoidal structure $\otimes$ (see monoidal categories), such that every object has left and right duals (it is rigid) and $\operatorname{End}(\mathbf{1}) = k$ .

Definition— fusion category and Frobenius–Perron dimension

A fusion category is a semisimple tensor category with finitely many isomorphism classes of simple objects $X_0 = \mathbf{1}, X_1, \dots, X_r$ . Its Grothendieck ring is a fusion ring: a ring free on the $[X_i]$ with $[X_i][X_j] = \sum_k N_{ij}^k [X_k]$ , $N_{ij}^k \in \N$ . The Frobenius–Perron dimension $\operatorname{FPdim}(X_i)$ is the largest real eigenvalue of the non-negative matrix of multiplication by $[X_i]$ ; $\operatorname{FPdim}$ is the unique ring homomorphism $K_0 \to \R$ that is positive on every $[X_i]$ , and $\operatorname{FPdim}(\mathcal{C}) = \sum_i \operatorname{FPdim}(X_i)^2$ .

Definition— module category

A (left) module category over a monoidal category $\mathcal{C}$ is a category $\mathcal{M}$ with an action bifunctor $\otimes : \mathcal{C} \times \mathcal{M} \to \mathcal{M}$ and natural isomorphisms $(X \otimes Y) \otimes M \cong X \otimes (Y \otimes M)$ and $\mathbf{1} \otimes M \cong M$ satisfying the pentagon and triangle axioms. It is the categorified notion of a module over a ring: $K_0(\mathcal{M})$ is a module over $K_0(\mathcal{C})$ .

Definition— sl₂-categorification (Chuang–Rouquier)

An $\mathfrak{sl}_2$ -categorification on a $k$ -linear abelian category $\mathcal{A}$ (with suitable finiteness) consists of exact functors $E, F : \mathcal{A} \to \mathcal{A}$ that are left and right adjoint to each other, such that $[E]$ and $[F]$ make $K_0(\mathcal{A}) \otimes \mathbb{Q}$ a locally finite representation of $\mathfrak{sl}_2$ in which the classes of simple objects are weight vectors, together with natural transformations $X \in \operatorname{End}(E)$ and $T \in \operatorname{End}(E^2)$ satisfying the relations of the degenerate affine Hecke algebra.

The extra natural transformations $X$ and $T$ are the essential, genuinely categorical part: an action of $\mathfrak{sl}_2$ on $K_0$ alone would not suffice for the theorems below.

Worked examplesH

Examples

Example— Finite sets categorify the natural numbers

The category $\mathbf{FinSet}$ has $\sqcup$ and $\times$ ; taking cardinalities gives $\N$ with $+$ and $\cdot$ . Commutativity, associativity and distributivity of arithmetic are shadows of specified bijections such as $A \times (B \sqcup C) \cong (A \times B) \sqcup (A \times C)$ , and these bijections satisfy coherence conditions of their own (they form a symmetric bimonoidal category). The numbers forget which bijection; the category remembers. For vector spaces, $\dim$ identifies $K_0(\mathrm{vect}_k)$ with $\Z$ .

Example— Rep(S₃)

Over $\mathbb{C}$ , $S_3$ has three irreducible representations: trivial $\mathbf{1}$ , sign $\varepsilon$ , and the $2$ -dimensional reflection representation $V$ . The fusion rules are

\varepsilon \otimes \varepsilon \cong \mathbf{1}, \qquad \varepsilon \otimes V \cong V, \qquad V \otimes V \cong \mathbf{1} \oplus \varepsilon \oplus V,

as the characters $(1, 1, 1)$ , $(1, -1, 1)$ , $(2, 0, -1)$ on the classes of $e$ , transpositions and $3$ -cycles confirm: $\chi_V^2 = (4, 0, 1) = \chi_{\mathbf 1} + \chi_\varepsilon + \chi_V$ . Frobenius–Perron dimensions are ordinary dimensions, and $\operatorname{FPdim}(\operatorname{Rep}(S_3)) = 1 + 1 + 4 = 6 = |S_3|$ .

Example— One fusion ring, several fusion categories

Pointed categories. For a finite group $A$ and a $3$ -cocycle $\omega \in Z^3(A, k^\times)$ , the category $\mathrm{Vec}_A^\omega$ of $A$ -graded vector spaces with associativity twisted by $\omega$ has Grothendieck ring $\Z[A]$ for every $\omega$ ; and $\mathrm{Vec}_A^\omega \simeq \mathrm{Vec}_A^{\omega'}$ only if $\omega'$ is cohomologous to the pull-back of $\omega$ by an automorphism of $A$ . For $A = \Z/2$ , $H^3(\Z/2, \mathbb{C}^\times) \cong \Z/2$ gives two inequivalent fusion categories with the same fusion ring.
$D_4$ and $Q_8$ . The dihedral group of order $8$ and the quaternion group have the same character table, hence isomorphic fusion rings; Tambara and Yamagami (1998) showed that $\operatorname{Rep}(D_4)$ and $\operatorname{Rep}(Q_8)$ are nevertheless inequivalent tensor categories.

Example— Fibonacci and Ising

The Fibonacci fusion ring has basis $\mathbf{1}, \tau$ with $\tau \otimes \tau = \mathbf{1} \oplus \tau$ ; it is the Grothendieck ring of fusion categories built from quantum groups at roots of unity (for instance the even part of $SU(2)$ at level $3$ ). The matrix of multiplication by $\tau$ is $\left(\begin{smallmatrix} 0 & 1 \\ 1 & 1 \end{smallmatrix}\right)$ , so $\operatorname{FPdim}(\tau) = \varphi = \tfrac{1 + \sqrt 5}{2}$ , the golden ratio. The Ising ring has $\sigma \otimes \sigma = \mathbf{1} \oplus \psi$ , $\psi \otimes \psi = \mathbf{1}$ , $\sigma \otimes \psi = \sigma$ , and $\operatorname{FPdim}(\sigma) = \sqrt{2}$ .

Example— Module categories over Vec_G

For a subgroup $H \le G$ , the category of vector spaces graded by the cosets $G/H$ is a module category over $\mathrm{Vec}_G$ (with $g$ acting by left translation of the grading), and twisting by a $2$ -cocycle $\psi \in Z^2(H, k^\times)$ gives more. Ostrik (2003) proved that every indecomposable semisimple module category over $\mathrm{Vec}_G$ is of this form $\mathcal{M}(H, \psi)$ , with $(H, \psi)$ determined up to conjugation and coboundaries — the categorified version of "a transitive $G$ -set is $G/H$ ".

Example— Categorifying a Hecke-type relation with symmetric groups

Let $\mathcal{A} = \bigoplus_{n \ge 0} \operatorname{Rep}_{\mathbb{C}}(S_n)$ , with $F = \operatorname{Ind}_{S_n}^{S_{n+1}}$ and $E = \operatorname{Res}^{S_{n+1}}_{S_n}$ , which are biadjoint. Mackey's formula gives an isomorphism of functors $E F \cong F E \oplus \mathrm{id}$ on each $\operatorname{Rep}(S_n)$ . On $K_0(\mathcal{A}) = \bigoplus_n R(S_n)$ , which is identified with the ring of symmetric functions, this is the Heisenberg-type relation $[E][F] - [F][E] = 1$ — a categorified commutation relation (Khovanov, 2014, studied the full Heisenberg categorification). In characteristic $p$ , splitting $E$ and $F$ according to eigenvalues of the Jucys–Murphy elements gives $i$ -restriction and $i$ -induction functors $E_i, F_i$ and a categorical action of $\widehat{\mathfrak{sl}}_p$ whose $K_0$ is the basic representation. For the Hecke algebras $H_n(q)$ over $\mathbb{C}$ at a primitive $e$ -th root of unity the same construction gives an action of $\widehat{\mathfrak{sl}}_e$ , and Ariki's theorem (1996, the Lascoux–Leclerc–Thibon conjecture) identifies their decomposition numbers with coefficients of Lusztig's canonical basis of the Fock space.

Example— The Khovanov cube of the Hopf link

Each crossing of a link diagram can be smoothed in two ways, $0$ and $1$ . Khovanov assigns to a smoothing with $c$ circles the graded vector space $V^{\otimes c}$ , with $V = \mathbb{Q}[x]/(x^2)$ , and to each edge of the cube of smoothings the multiplication $m : V \otimes V \to V$ (two circles merge) or the comultiplication $\Delta : V \to V \otimes V$ (a circle splits). For the Hopf link the smoothings $00, 01, 10, 11$ have $2, 1, 1, 2$ circles:

The cube of smoothings of the Hopf link. Collapsing the columns gives a chain complex V⊗V → V ⊕ V → V⊗V (with degree shifts); the sign makes the square anticommute, so d∘d = 0.

The graded Euler characteristic of this complex, with the standard normalisation shifts, is the unnormalised Jones polynomial; its homology is the Khovanov homology. The computation section does this for the trefoil and the figure-eight knot.

ProofH

Central results

Theorem— Proposition 1 — dimension decategorifies vect

If $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ is a short exact sequence of finite-dimensional vector spaces, then $\dim B = \dim A + \dim C$ . Hence $[V] \mapsto \dim V$ is a well-defined isomorphism $K_0(\mathrm{vect}_k) \cong \Z$ .

Proof

Rank–nullity for $g$ gives $\dim B = \dim \operatorname{im} g + \dim \ker g$ . Exactness says $\operatorname{im} g = C$ and $\ker g = \operatorname{im} f \cong A$ ( $f$ injective). So $\dim$ respects the defining relations and induces a homomorphism $K_0 \to \Z$ . It is surjective ( $[k] \mapsto 1$ ) and injective, since $V \cong k^{\dim V}$ gives $[V] = \dim V \cdot [k]$ .

∎

Theorem— Proposition 2 — homology categorifies the Euler characteristic

For a bounded complex $C$ of finite-dimensional vector spaces, $\sum_i (-1)^i \dim C^i = \sum_i (-1)^i \dim H^i(C)$ .

Proof

Write $Z^i = \ker d^i$ and $B^i = \operatorname{im} d^{i-1}$ . Rank–nullity gives $\dim C^i = \dim Z^i + \dim B^{i+1}$ , and $\dim H^i = \dim Z^i - \dim B^i$ . Hence $\sum_i (-1)^i \dim C^i = \sum_i (-1)^i (\dim H^i + \dim B^i + \dim B^{i+1})$ , and the boundary terms cancel in pairs. In the language of Grothendieck groups: $[C] := \sum (-1)^i [C^i] = \sum (-1)^i [H^i]$ in $K_0(\mathrm{vect}_k)$ , which is the statement that $K_0$ of the derived category $D^b(\mathrm{vect}_k)$ is again $\Z$ .

∎

Theorem— Proposition 3 — the Grothendieck ring of Rep(G)

Let $G$ be a finite group and $k = \mathbb{C}$ . Then $K_0(\operatorname{Rep}(G))$ is the free abelian group on the classes of the irreducible representations; with $[V][W] = [V \otimes W]$ it is the representation ring $R(G)$ , and the character map $[V] \mapsto \chi_V$ is an injective ring homomorphism into class functions. Moreover $\operatorname{Rep}(G)$ is a fusion category with $\operatorname{FPdim}(V) = \dim V$ and $\operatorname{FPdim}(\operatorname{Rep}(G)) = |G|$ .

Proof

By Maschke's theorem every short exact sequence splits and every representation is a direct sum of irreducibles $V_1, \dots, V_r$ , uniquely up to multiplicities, so $K_0 = \bigoplus_i \Z [V_i]$ . The character is additive on direct sums, multiplicative on tensor products ( $\chi_{V \otimes W}(g) = \operatorname{tr}(\rho_V(g) \otimes \rho_W(g)) = \chi_V(g)\chi_W(g)$ ), constant on isomorphism classes, and sends $\mathbf{1}$ to the constant function $1$ : a ring homomorphism. It is injective because the irreducible characters are orthonormal for $\langle \chi, \psi \rangle = \frac{1}{|G|} \sum_g \chi(g) \overline{\psi(g)}$ , hence linearly independent (see representation theory of finite groups). Duals exist ( $V^*$ with $g$ acting by the inverse transpose), $\operatorname{End}(\mathbf{1}) = \mathbb{C}$ , and there are finitely many simples: $\operatorname{Rep}(G)$ is fusion. Finally $\dim = \chi(e)$ is a ring homomorphism $K_0 \to \Z \subset \R$ , positive on each $[V_i]$ , so by uniqueness it is $\operatorname{FPdim}$ , and $\sum_i (\dim V_i)^2 = |G|$ .

∎

Theorem— Theorem 4 — the Fibonacci rule is not Rep(G)

There is no finite group $G$ (and no semisimple Hopf algebra $H$ ) whose representation category has Grothendieck ring isomorphic, as a ring with basis, to the Fibonacci fusion ring. In fact the Fibonacci ring has no ring homomorphism to $\Z$ at all.

Proof

A ring homomorphism $d$ from the Fibonacci ring to $\Z$ sends $\tau$ to an integer $n$ with $n^2 = d(\tau^2) = d(1 + \tau) = 1 + n$ , i.e. $n(n - 1) = 1$ . But a product of two consecutive integers is never $1$ : it is $0$ for $n \in \{0, 1\}$ and at least $2$ otherwise. So there is no such $n$ . But for $\operatorname{Rep}(G)$ (or $\operatorname{Rep}(H)$ ), taking the dimension of the underlying vector space is a ring homomorphism $K_0 \to \Z$ , because the forgetful functor to vector spaces preserves $\oplus$ and $\otimes$ .

∎

The same argument shows that a fusion category with a non-integer Frobenius–Perron dimension admits no fibre functor (no monoidal functor to vector spaces), so it cannot be reconstructed as the representations of a group or Hopf algebra. Fusion categories are strictly more general than $\operatorname{Rep}(G)$ .

Theorem— Theorem 5 — Khovanov (2000)

For an oriented link $L$ there are bigraded homology groups $\mathit{Kh}^{i,j}(L)$ , computed from any diagram by the cube of smoothings, which are invariant under Reidemeister moves, and whose graded Euler characteristic is the unnormalised Jones polynomial: $\sum_{i,j} (-1)^i q^j \dim \mathit{Kh}^{i,j}(L) = (q + q^{-1}) J_L$ (with Bar-Natan's normalisation of $J_L$ ).

Khovanov's proof of invariance is a direct verification for the three Reidemeister moves (see Bar-Natan, 2002, for an accessible account). Khovanov homology is functorial for link cobordisms up to sign (Jacobsson 2004; Khovanov 2006; Bar-Natan 2005), with the sign ambiguity resolved by Clark, Morrison and Walker (2009). It is strictly stronger than the Jones polynomial: the figure-eight knot has $\hat J = q^{-5} + q^{5}$ but $\mathit{Kh}$ of total rank $6$ (computed below).

Theorem— Theorem 6 — Chuang–Rouquier (2008)

An $\mathfrak{sl}_2$ -categorification on $\mathcal{A}$ lifts the action of the simple reflection of the Weyl group on $K_0(\mathcal{A})$ to a derived self-equivalence: the weight spaces $\mathcal{A}_\lambda$ and $\mathcal{A}_{-\lambda}$ are derived equivalent. Applied to blocks of symmetric groups in characteristic $p$ with the $i$ -induction and $i$ -restriction functors, this shows that blocks with the same weight are derived equivalent; together with the work of Chuang and Kessar (2002), it proves Broué's abelian defect group conjecture for symmetric groups.

We do not prove Theorems 5 and 6 here; see the reading list.

ExplanationH

The landscape: what is known, what is open

Established results. Beyond the theorems above:

Positivity by categorification. Kazhdan–Lusztig positivity for all Coxeter groups (Elias–Williamson, 2014, via the Hodge theory of Soergel bimodules); positivity of Lusztig's canonical bases (via perverse sheaves on quiver representation spaces, see quiver representations).
Categorified quantum groups. Khovanov–Lauda and Rouquier (2008–2011) defined 2-categories whose Grothendieck groups are quantum groups, with KLR (quiver Hecke) algebras categorifying their positive parts.
Knot homology. Khovanov homology detects the unknot (Kronheimer–Mrowka, 2011). Rasmussen's $s$ -invariant (2010) gives a combinatorial proof of the Milnor conjecture on the slice genus of torus knots, first proved by Kronheimer–Mrowka with gauge theory; Piccirillo (2020) used it to show that the Conway knot is not slice.
Structure of fusion categories. Etingof, Nikshych and Ostrik (2005): fusion categories over $\mathbb{C}$ admit no non-trivial deformations ("Ocneanu rigidity"), so a fusion ring has only finitely many categorifications up to equivalence; FP dimensions are algebraic integers; $\operatorname{FPdim}(\mathcal{C})$ is divisible, in the ring of algebraic integers, by $\operatorname{FPdim}$ of any fusion subcategory. Ostrik (2003) showed that rank- $2$ fusion categories have the fusion rings of $\Z/2$ or Fibonacci.
Topology from fusion categories. Spherical fusion categories give the Turaev–Viro invariants of $3$ -manifolds (1992), and modular tensor categories give Reshetikhin–Turaev invariants (1991) and $3$ -dimensional topological quantum field theories.

Open problems (well documented; consult recent surveys for progress):

Does the Jones polynomial detect the unknot? Open, although Khovanov homology does.
Etingof–Nikshych–Ostrik (2011) asked whether every fusion category of integer Frobenius–Perron dimension is weakly group-theoretical. Open.
The Broué abelian defect group conjecture for finite groups in general (proved for symmetric groups by Theorem 6, and in other families). Open.
Do the "exotic" Haagerup fusion categories (from Haagerup's 1994 subfactor) arise from a conformal field theory? Open; there is numerical and structural evidence (Evans–Gannon, 2011).

Formal mathematicsF

Categorification in Lean and Mathlib

Mathlib has finite types and their cardinalities, finite-dimensional linear algebra, and the category FDRep k G of finite-dimensional representations with its symmetric monoidal and rigid structures and characters. It does not yet have fusion categories, module categories or categorical actions. We formalise the decategorification statements of the lesson.

Finite sets categorify $\N$ : cardinality turns $\sqcup$ and $\times$ into $+$ and $\cdot$ , and the distributive law is the shadow of a bijection.

card_sumLean 4Applies a library theoremFormally verified

/-- Decategorifying the disjoint union: `|A ⊔ B| = |A| + |B|`. This applies Mathlib's
`Fintype.card_sum`. -/
theorem card_sum (A B : Type*) [Fintype A] [Fintype B] :
    Fintype.card (A ⊕ B) = Fintype.card A + Fintype.card B :=
  Fintype.card_sum