Minimal faithful linear representations (awaiting review)

A group $G$ can always be represented as a matrix group, but the degree of the matrix group is constrained both by $G$ and the ring $R$ from which the matrix entries are drawn. The linear degree $\operatorname{lin}_R(G)$ is the minimal degree $n$ of a faithful linear representation $\rho : G \hookrightarrow \GL_n(R)$, and the irreducible degree $\operatorname{irr}_R(G)$ is the minimal degree $n$ of a faithful irreducible representation $\rho: G \hookrightarrow \GL_n(R)$. Note that, while a faithful linear representation always exists, there may be no faithful irreducible representations of $G$ (in which case we conventionally set $\operatorname{irr}_R(G) = -1$).

For fields $R$ of characteristic 0, these values can be characterized in terms of the character table of $G$. A character $\chi$ of degree $n$ is faithful if the only $g \in G$ with $\chi(g) = n$ is $g=1$. The value $\operatorname{irr}_{\mathbb{C}}(G)$ can thus be read off of the character table, but computing $\operatorname{lin}_{\mathbb{C}}(G)$ requires finding linear combinations of characters that are faithful.

For fields $k$ that are not algebraically closed, summing over Galois orbits of complex characters yields characters with values in $k$, and the resulting characters will be faithful if and only if each of the summands is. However, the Schur index now plays a role, since multiple copies of the given $k$-values character may be needed in order to produce an actual representation with matrix entries in $k$. Both $\operatorname{lin}_R(G)$ and $\operatorname{irr}_R(G)$ are defined in terms of the matrix degree of a representation, which may be larger than the degree of a minimal faithful $k$-valued character.