  We continue the study of low dimensional linear representations of mapping
class groups of surfaces initiated by Franks--Handel and Korkmaz. We consider
$(2g+1)$-dimensional complex linear representations of the pure mapping class
groups of compact orientable surfaces of genus $g$. We give a complete
classification of such representations for $g \geq 7$ up to conjugation, in
terms of certain twisted $1$-cohomology groups of the mapping class groups. A
new ingredient is to use the computation of a related twisted $1$-cohomology
group by Morita. The classification result implies in particular that there are
no irreducible linear representations of dimension $2g+1$ for $g \geq 7$, which
marks a contrast with the case $g=2$.
