13 problems

For each of the 23 umbral cases, let $G^X$ be the corresponding finite group, let $m$ be its associated positive integer, and let $I^X\subset\{1,2,\dots,m-1\}$ be the specified sub…

Let $X$ be a Niemeier root system and let $g\in G^X$. Let $\check{H}^X_g$ denote the vector formed by the components of $H^X_g$ indexed by $0<r<m$, and let…

Umbral moonshine module conjecture. There is a naturally defined module $\check{K}^X$ such that, for $g\in G^X$ and $r\in I^X$,

Let $X$ be a Niemeier root system with Coxeter number $m$. A representation is a doublet if it is the direct sum of two copies of one representation, and a sextet if it is the dire…

Let $X$ be a Niemeier root system with Coxeter number $m$, and let $D$ be a discriminant of $H^X$. Dual-pair conjecture. If … for some integer $\lambda$, then the representation…

Let $X$ be a Niemeier root system with Coxeter number $m$, let $G^X$ be its associated group, and let $S^X_g$ be the attached cusp form for $g\in G^X$. For $g\in G^X$, let $n_g$ be…

Factoring-through conjecture. The $\mathbb{Q}$-graded $G^X$-module $K^X_r$ is a faithful representation of $G^X$ when $r\equiv0\pmod c$, and factors through $\bar G^X$ otherwise. T…

Let $\Lambda=\{2,3,4,5,7,13\}$ and let $K^{(\ell)}_{r,-D/4\ell}$ be the corresponding representation of $G^{(\ell)}$. Say that a $G$-module is a doublet if it is isomorphic to the…

Let $H^{(\ell)}$ be an umbral vector-valued mock modular form, let $D$ be a discriminant of $H^{(\ell)}$, and let $K^{(\ell)}_{r,-D/4\ell}$ be the corresponding representation of…

Let $\Lambda=\{2,3,4,5,7,13\}$, let $G^{(\ell)}$ be the corresponding umbral group, and let $H^{(\ell)}_g$ be its vector-valued mock modular McKay–Thompson series. For each…

Let $\Lambda=\{2,3,4,5,7,13\}$, let $g\in G^{(\ell)}$, and let $H^{(\ell)}_g=(H^{(\ell)}_{g,r})$ be the vector-valued mock modular form obtained from the graded supertrace function…

Let $\Lambda=\{2,3,4,5,7,13\}$, let $G^{(\ell)}$ be the corresponding umbral group, and let $H^{(\ell)}_g$ be its vector-valued mock modular McKay–Thompson series. For…

Umbral moonshine module conjecture. For every $\ell\in\Lambda$, such a $G^{(\ell)}$-module exists and satisfies