  The still-unsolved problem of determining the set of eigenvalues realized by
$n$-by-$n$ doubly stochastic matrices, those matrices with row sums and column
sums equal to $1$, has attracted much attention in the last century. This
problem is somewhat algebraic in nature, due to a result of Birkhoff
demonstrating that the set of doubly stochastic matrices is the convex hull of
the permutation matrices. Here we are interested in a general matrix group $G
\subseteq GL_n(\mathbb{C})$ and the hull spectrum $\text{HS}(G)$ of eigenvalues
realized by convex combinations of elements of $G$. We show that hull spectra
of matrix groups share many nice properties. Moreover, we give bounds on the
hull spectra of matrix groups, determine $\text{HS}(G)$ exactly for important
classes of matrix groups, and study the hull spectra of representations of
abstract groups.
