112 problems

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

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…

Let $Sp^{\Sigma}$ be the category of symmetric spectra and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Let $LSp^{\Sigma}$ be the tensor Segal categor…

Let $Sp^{\Sigma}$ be the category of symmetric spectra with its positive model structure, and let $Comm(Sp^{\Sigma})$ be the model category of commutative ring spectra. Write…

Let ${\mathcal C}$ be a finite tensor category, let $V\in{\mathcal C}$, let $V^{**}$ denote the double dual of $V$, and let $L_\rho$ be the distinguished invertible object of…

Circulator trace conjecture. In the case of stable Gelfand–Kazhdan categories,

Let $\mathcal{C}$ be a finitely tensor-generated pre-Tannakian category of moderate growth. Write $\mathcal{M}\mathcal{Z}(\mathcal{C})$ for its Müger center, and let…

Let $\mathfrak g$ be a semisimple Lie algebra with weight and root lattices $P$ and $Q$, respectively. For $q\in\mathbb{C}$, let $U_q(\mathfrak g)\mathbf{-mod}$ be the category of…

Crystal Temperley–Lieb center conjecture. The canonical functor $G$ is a monoidal equivalence. This conjecture asserts that the only objects in the Drinfeld center are those induce…

Let $k$ be a field, let $\mathbf Q(\mathbb Z^3)$ be the QCA space in dimension three, and let $\mathrm{Witt}_k$ denote the Witt group of braided fusion $k$-linear categories. The Q…

Let $\mathfrak{g}$ be a Lie algebra, let $e\in\mathfrak{g}$ be distinguished nilpotent, and let $Q=Q(e)$ be the finite centralizer of an $sl_2$-triple associated with $e$ in the si…

Let $\mathcal{C}$ be a unimodular finite braided tensor category, and let $\operatorname{Ver}^{\zeta^{1/2}}_{p^{(n)}}$ denote a mixed higher Verlinde category. Regard these categor…

Logarithmic Kazhdan–Lusztig conjecture. In good cases, there is an equivalence of braided tensor categories

Generalized Benson conjecture. If $(\mathcal{C},G')$ is an induced $p'$-symmetry of $V$, then every prime dividing the order of $G'$ is less than $p$.

Generalized Benson conjecture. Every prime dividing the order of $G'$ is less than $p$.

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 $\Bbbk$;…

Fix a finite field $\mathbf{F}$ of odd cardinality $q$. For $n\geq 0$, let $H_n=\operatorname{\mathbf{Sp}}_{2n}(\mathbf{F})$, let $k_n$ be an algebraically closed field, and let…

Tensor-product equivalence conjecture. The map above induces an equivalence of monoidal categories

Dimension conjecture. Every T-prime ideal admits a dimension.

Tensor-subcategory conjecture. If $\mathcal{C}$ is Frobenius exact and the annihilator map

Annihilator-map conjecture. The annihilator map

Let $\mathcal{E}$ be a finite symmetric tensor category. Let $\mathrm{Mor}_2^{\mathrm{fin}}(\mathbf{Pr}_{\mathcal{E}})$ be the symmetric monoidal sub-4-category whose objects are f…