227 problems

Benson's conjecture. All indecomposable summands of $V\otimes V^*$ have dimension divisible by $2$, except for the single summand isomorphic to the trivial representation $\Bbbk$;…

Etingof's pivotal structure conjecture. Every fusion category admits a pivotal structure.

Let $\widehat{\mathscr{W}}_{3,2}$ be the module category of the $\mathfrak{sl}_3$ minimal model $\mathsf{A}_2(3,2)$. A rigid braided tensor category is a braided tensor category in…

Braided tensor structure conjecture. The quotient category $\mathcal{C}_{p_+,p_-}/\mathcal{S}$ has the structure of a braided tensor category.

Conjecture on $L_{k_m}(D_4)$. For each $m\geq1$, $L_{k_m}(D_4)$ is a quasi-lisse vertex algebra, it has a unique irreducible ordinary module, and $KL_{k_m}(D_4)$ is a semisimple br…

Let $W$ be a vertex operator algebra containing Heisenberg fields $\alpha_i$, and let $S_i=\operatorname{Res}(e^{\alpha_i})$. Assume that $W$ decomposes into integer eigenvalues un…

Let $k$ be a perfect field, and let $\mathcal{A}$ be a finite braided tensor category over $k$. The symmetric center of $\mathcal{A}$ is finite semisimple precisely when…

Let $\boldsymbol{k}$ be an algebraically closed field of characteristic $p>0$. A tensor category is incompressible if every tensor functor out of it is an embedding of a tensor sub…

Let $\mathcal{W}_p$ be the triplet VOA and let $\overline{U}_q(\mathfrak{sl}_2)$ be the restricted quantum group at a $2p$-th root of unity. FGST equivalence conjecture. The repres…

For each integer $p\,{\geqslant}\, 2$, let $\mathfrak{C}_p$ be the category of representations of the triplet $\boldsymbol{\mathcal{W}}(p)$ algebra whose adjoint $L_0$-eigenspaces…

Higher Verlinde fibre conjecture. All symmetric tensor categories of moderate growth fibre over $\mathsf{Ver}_{p^\infty}$. If so, the study of such categories would reduce, via the…

Let $\mathcal{E}$ be a symmetric fusion category and let $\mathcal{C}$ be a non-degenerate braided fusion category over $\mathcal{E}$. For a fusion category $\mathcal{D}$ containin…

BEO's classification conjecture. Every symmetric tensor category of moderate growth over $\mathbf{k}$ is a representation category over the symmetric tensor category…

Let $\mathcal{C}$ be a finite pointed tensor category, meaning that every simple object of $\mathcal{C}$ is invertible. An object has length given by the relevant tensor-generation…

Let $k$ be an algebraically closed field of positive characteristic $p$, and let $\operatorname{Ver}_{p^n}$ denote the higher Verlinde categories, including…

Let $p$ be a positive integer and let $\mathrm{Ver}_p$ denote the Verlinde category. A tensor category has subexponential growth if its growth is subexponential in the sense used i…

Let $\mathcal{C}$ be a finite tensor category over an algebraically closed field $\mathbf{k}$ of characteristic $q\ge 0$, with Frobenius–Perron dimension $p>2$ prime. A tensor cate…

Tensor-product conjecture. The pair $(\widetilde M,\widetilde I(\cdot,z))=(M,Y_M(\Delta(z)\cdot,z))$ is a tensor product of $(M,\widetilde V)$ in the sense of the cited tensor-prod…

Let ${\cal SW}(V)$ be the Schur–Weyl category, let $V_{\lambda}$ denote its distinguished objects, and let ${\bf g}=sl(V_S)$ act on these objects. A linear map between two objects…

Tensor-functor realization conjecture. For every prime $p$ there exists a tensor functor

The modularity conjecture. The subcategory ${\mathcal M}_e$ is closed under convolution and hence is a monoidal category with unit $e$ if the preceding perversity conjecture holds;…

Let $\mathcal{M}$ be a category of finite-dimensional motives over a field $k$, and let $M\in\mathcal{M}$. Write $\det(M)$ for the determinant of $M$, let $\mathbb{L}$ denote the L…

Let ${\mathcal C}_\zeta^{{\underline c}_f}$ be the monoidal category associated with a finite two-sided cell ${\underline c}_f\subset W_f(\zeta)$, and let…

Let ${\underline c}_f\subset W_f(\zeta)$ be a two-sided cell whose corresponding family is not exceptional. Let ${\mathcal G}_{{\underline c}_f}$ be Lusztig's associated finite gro…

Rigidity conjecture. (i) Any fusion ring has at most finitely many realizations over ${\bf k}$, up to equivalence (possibly none). (ii) The number of tensor functors between two fi…