The Lumenary · Research pathways

Categorical representation theory

Name: ______________________

Date: ______________________

You will learn to

Explain categorification and decategorification through examples — natural numbers and finite sets, Euler characteristic and homology, the Jones polynomial and Khovanov homology
Define Grothendieck groups, tensor and fusion categories, Frobenius–Perron dimension and module categories, with $\operatorname{Rep}(G)$ as the guiding example
Prove that dimension and characters decategorify $\operatorname{Rep}(G)$ , and that the Fibonacci fusion rule is not the Grothendieck ring of any $\operatorname{Rep}(G)$
Describe categorical actions, including Chuang–Rouquier $\mathfrak{sl}_2$ -categorification, and separate established results from open problems

1.
In $\operatorname{Rep}(S_3)$ over $\mathbb{C}$ , with $V$ the $2$ -dimensional irreducible representation, how many copies of $V$ occur in $V \otimes V \otimes V$ ?
2.
What is $\operatorname{FPdim}(\operatorname{Rep}(D_4)) = \sum_i \operatorname{FPdim}(X_i)^2$ , the sum over the simple objects of the representation category of the dihedral group of order $8$ ?
3.
Which of these statements are true? (Choose all that apply.)
- A.The graded Euler characteristic of Khovanov homology is the unnormalised Jones polynomial
- B. $\dim$ induces an isomorphism $K_0(\mathrm{vect}_k) \cong \Z$
- C.The Grothendieck ring of $\mathrm{Vec}_A^{\omega}$ depends on the $3$ -cocycle $\omega$
- D.The Betti numbers of a space determine its integral homology groups
4.
How many fusion categories over $\mathbb{C}$ , up to monoidal equivalence, have Grothendieck ring $\Z[\Z/2]$ (two simple objects $\mathbf{1}, g$ with $g \otimes g \cong \mathbf{1}$ )?
5.
The proof that the Fibonacci fusion ring is not $K_0(\operatorname{Rep}(G))$ uses one property of $\operatorname{Rep}(G)$ . Which?
- A.Dimension of the underlying vector space is a ring homomorphism $K_0(\operatorname{Rep}(G)) \to \Z$
- B. $\operatorname{Rep}(G)$ is braided
- C. $\operatorname{Rep}(G)$ has finitely many simple objects
- D.Characters are constant on conjugacy classes
6.
The lesson's experiment finds, for the figure-eight knot, $\hat J = q^{-5} + q^{5}$ , while $\mathit{Kh}$ has six one-dimensional summands in bidegrees $(t, q) = (-2, -5), (-1, -1), (0, -1), (0, 1), (1, 1), (2, 5)$ . What does this show?
- A.Pairs of summands in adjacent homological degrees with the same $q$ -degree cancel in the Euler characteristic; homology keeps them
- B.The computation must be wrong, since homology and Euler characteristic should have the same number of terms
- C.Khovanov homology of the figure-eight knot is determined by its Jones polynomial
- D.The figure-eight knot is the unknot
7.
Which information about a fusion category $\mathcal{C}$ is lost when passing to its Grothendieck ring (as a ring with its basis of simple objects)? (Choose all that apply.)
- A.The associativity constraint (e.g. the class of $\omega$ in $\mathrm{Vec}_A^\omega$ )
- B.For $\mathcal{C} = \operatorname{Rep}(G)$ : the group $G$ up to isomorphism
- C.The number of simple objects
- D.The Frobenius–Perron dimensions of the simple objects
8.
Which Lean statement formalises “dimension is additive on short exact sequences $0 \to A \to B \to C \to 0$ ”? (Here f : A →ₗ[k] B, g : B →ₗ[k] C, and B is finite-dimensional.)
- A.(hf : Function.Injective f) (hg : Function.Surjective g) (hexact : LinearMap.range f = LinearMap.ker g) : finrank k B = finrank k A + finrank k C
- B.(hexact : LinearMap.range f = LinearMap.ker g) : finrank k B = finrank k A + finrank k C
- C.(hf : Function.Injective f) (hg : Function.Surjective g) : finrank k B = finrank k A + finrank k C
- D.finrank k (A × C) = finrank k A + finrank k C
9.
Write fusion_coefficients(sizes, chars) that takes the class sizes of a finite group and a list of its (real-valued) irreducible characters, each a list of values on the classes, and returns the nested list N[i][j][k] $= \langle \chi_i \chi_j, \chi_k \rangle = \frac{1}{|G|} \sum_c |c|\, \chi_i(c) \chi_j(c) \chi_k(c)$ as integers.
Computational — Practice

Answer key — Categorical representation theory

1.
$V^{\otimes 3} = V \otimes (\mathbf{1} \oplus \varepsilon \oplus V) = V \oplus V \oplus (\mathbf{1} \oplus \varepsilon \oplus V) = \mathbf{1} \oplus \varepsilon \oplus 3V$ . Check with dimensions: $8 = 1 + 1 + 3 \cdot 2$ . Check with characters: $\chi_V^3 = (8, 0, -1)$ and $(1, 1, 1) + (1, -1, 1) + 3(2, 0, -1) = (8, 0, -1)$ .
2.
$\operatorname{FPdim}$ equals $\dim$ in $\operatorname{Rep}(G)$ (Proposition 3), so $\operatorname{FPdim}(\operatorname{Rep}(D_4)) = 4 \cdot 1^2 + 2^2 = 8 = |D_4|$ . The same holds for $Q_8$ , which has the same fusion ring.
3.
(a) and (b). Statement (c) fails because $K_0$ forgets the associativity constraint, and (d) because Betti numbers forget torsion (and, more importantly, maps).
4.
Such a category is pointed, hence $\mathrm{Vec}_{\Z/2}^\omega$ for $\omega \in H^3(\Z/2, \mathbb{C}^\times) \cong \Z/2$ ; since $\operatorname{Aut}(\Z/2)$ is trivial, the two classes give two inequivalent categories with the same fusion ring. One fusion ring, two categorifications.
5.
(a). Any ring homomorphism $d$ to $\Z$ would satisfy $d(\tau)^2 = 1 + d(\tau)$ , which has no integer solution (Lean: fibonacci_no_integer_dimension). For $\operatorname{Rep}(G)$ , $\dim$ is such a homomorphism, so the rings cannot be isomorphic.
6.
(a). $\sum (-1)^t q^j = q^{-5} - q^{-1} + q^{-1} + q - q + q^{5} = q^{-5} + q^{5}$ : two cancelling pairs disappear from the Euler characteristic but not from homology. (Also, since the unknot's Khovanov homology has rank $2$ and Khovanov homology detects the unknot, rank $6$ shows the knot is non-trivial.)
7.
(a) and (b). The based ring knows the simples and the $N_{ij}^k$ , hence the rank and the FP dimensions (Perron–Frobenius eigenvalues of the fusion matrices). It forgets associators and, for representation categories, which group they came from — and even the tensor category itself (Tambara–Yamagami).
8.
(a). A short exact sequence $0 \to A \to B \to C \to 0$ means: $f$ injective, $g$ surjective, $\operatorname{im} f = \ker g$ . Dropping any of them makes the statement false, as the counterexamples show; (d) is the special case of a split sequence.

def fusion_coefficients(sizes, chars):
    order = sum(sizes)
    r = len(chars)
    out = []
    for i in range(r):
        row = []
        for j in range(r):
            entry = []
            for k in range(r):
                total = sum(s * a * b * c for s, a, b, c in
                            zip(sizes, chars[i], chars[j], chars[k]))
                entry.append(total // order)
            row.append(entry)
        out.append(row)
    return out

For $S_3$ this gives $V \otimes V = \mathbf{1} \oplus \varepsilon \oplus V$ . That the averages are non-negative integers is a theorem (they are multiplicities, by orthonormality of characters); the computation is the decategorified shadow of decomposing $V_i \otimes V_j$ in $\operatorname{Rep}(G)$ . For complex-valued characters one would use $\overline{\chi_k}$ .