11 problems

Let $\mathcal C$ be a near-group category of type $\mathbb Z/4\mathbb Ztimes \mathbb Z/4\mathbb Z+16$, provided that such a category exists, and let $Z(\mathcal C)$ denote its Drin…

For $n\geq 1$, define … where $0_n=(0,0,\ldots,0)\in\mathbb{F}_2^n$ and $I_n\in SL_n(\mathbb{F}_2)$ is the identity matrix. Then $G_n\cong\mathbb{F}_2^{2n}\rtimes S_3$. Let…

Izumi's conjecture. All six pointed categories have the group $\mathbb{F}_2^{2n}\rtimes SL(2,\mathbb{F}_2)$, with the six different third cohomology classes coming from…

Let $G_o$ be an odd metric group and let $n\geq1$. Cyclic 2-group near-group conjecture. The pair of tensor-product modular data displayed in the source is the modular data of the…

Let the fifth family of potential modular data be obtained from the fourth family by switching the roles of $G$ and $\Gamma$. Fifth-family realization conjecture. Examples of this…

Let $A$ be an odd finite abelian group and let $G=\mathbb{Z}_{2^n}\times A$ with $n\geq1$. Inverse-involution near-group conjecture. The constructed modular data are the modular da…

Let $A$ be a finite abelian group, let $G=A\times A$, and let $\langle\langle\cdot,\cdot\rangle\rangle$ be a non-degenerate symmetric bicharacter on $A$. Define…

Divisibility conjecture. The order $|G_\rho|$ divides $K$. This conjecture proposes an arithmetic constraint on the multiplicities in generalized near-group fusion categories; the…

Let $p$ be prime, and let $\operatorname{Vec}_{\mathbb{Z}_p}$ be the fusion category of finite-dimensional $\mathbb{Z}_p$-graded vector spaces. Consider the algebra…

Parameterization conjecture. The six elements of $H^3(SL(2,2),\mathbb{T})$ parameterize six different $C^*$ near-group categories $\mathcal{C}_{G,\zeta}$ with $|G|=2^{2k+1}$.

Let $G$ be a finite abelian group of odd order, let $q$ be a quadratic form on $G$, and let $\mathcal{NG}(G,q,b,c)$ be the corresponding near-group fusion category. Let $E_j$ be th…