Maybe I am misinterpreting something, because ccording to my experiments, this function is neither convex nor concave.
The following is a counterexample to convexity.
\begin{equation*} A=\begin{pmatrix}10 &8\\ 2 & 8\end{pmatrix},\qquad B=\begin{pmatrix} 8 & 8\\ 8 & 6\end{pmatrix}. \end{equation*} Also, let $C=(A+B)/2$. For this choice, we have
\begin{equation*} v(A) = (.9315, .3637),\quad v(B)=(.7497,.6618),\quad v(C)=(.8287, .5597). \end{equation*} But $\|v(C)\|_1 > 0.5\|v(A)\|_1 + 0.5\|v(B)\|_1$ (notice all there vectors have unit 2-norm as required).
A similar counterexample to potential concavity is also easy to find.