  Let $k$ and $n$ be positive coprime integers with $k<n$. Let $T$ denote the
subgroup of diagonal matrices in $SL(n,\mathbb{C})$. We study the GIT quotient
of Richardson varieties $X^v_w$ in the Grassmannian $\mathrm{Gr}_{k,n}$ by $T$
with respect to a $T$-linearised line bundle $\cal{L}$ corresponding to the
Pl\"{u}cker embedding. We give necessary and sufficient combinatorial
conditions for the quotient variety $T \backslash\mkern-6mu\backslash
(X_w^v)^{ss}_T({\cal L})$ to be smooth.
