Is this "coincidence" about representations of the Monster actually a coincidence? I know that the Monster simple group's lowest dimension faithful representation (which is in characteristic $2$) has dimension $196882$ and that its lowest dimension faithful representation in characteristic $0$ has dimension $196883$. Is there any simple explanation for the fact the dimension of the lowest dimension faithful representation has dimension one less than lowest dimension faithful representation in characteristic $0$? Are these reasons also valid for other simple groups, like $A_5$ and the Baby Monster groups? Are there representations of dimension $196882$ in other characteristics?
 A: I don't know whether this is what happens for the Monster specifically, but this is the sort of thing that could happen: it's known that every complex irreducible representation of a finite group is in fact defined over the ring of integers of a number field. So the Monster's $196883$-dimensional irreducible representation over $\mathbb{C}$ is defined over $\mathcal{O}_K$ for some number field $K$, and taking $P$ to be some prime ideal lying over $(2)$ in this ring of integers, we can reduce the representation over $\mathcal{O}_K$ mod $P$ to get a representation over $\mathbb{F}_{2^k}$ for some $k$.
Now, if it happens both that $k = 1$ and that the reduced representation $\bmod P$ is not irreducible but contains a $1$-dimensional invariant subspace, then we could quotient by this subspace and get a $196882$-dimensional representation over $\mathbb{F}_2$, which could be the $196882$-dimensional irreducible over $\mathbb{F}_2$.
Again, I don't know if this actually happens, but something like this happening is plausible a priori and it would give us a way of relating these two representations. We can probably find examples of this happening with smaller groups.
A: Griess and Smith prove in
Griess, Robert L. jun.; Smith, Stephen D., Minimal dimensions for modular representations of the Monster, Commun. Algebra 22, No. 15, 6279-6294 (1994). ZBL0820.20021.
that the $196883$-dimensional characteristic zero representation of the monster has a $p$-modular reduction that is irreducible for $p>3$ and has a $196882$-dimensional (irreducible) composition factor and a one-dimensional composition factor for $p=2$ and $p=3$.
