  In heterotic string theories consistency requires the introduction of a
non-trivial vector bundle. This bundle breaks the original ten-dimensional
gauge groups $\text{E}_8\times\text{E}_8$ or $\text{SO}(32)$ for the
supersymmetric heterotic string theories and $\text{SO}(16) \times
\text{SO}(16)$ for the non-supersymmetric tachyon-free theory to smaller
subgroups. A vast number of MSSM-like models have been constructed up to now,
most of which describe the vector bundle as a sum of line bundles. However,
there are several different ways of describing these line bundles and their
embedding in the ten-dimensional gauge group. We recall and extend these
different descriptions and explain how they can be translated into each other.
