Derivations preserving a monomial ideal

Let $I$ be a monomial ideal in a polynomial ring $\mathbf{A}=\mathbf{k}[x_1,\ldots, x_n]$ over a field $\mathbf{k}$ of characteristic 0, $T_{\mathbf{A}/\mathbf{k}} (I)$ be the module of $I$-preserving $\mathbf{k}$-derivations on $\mathbf{A}$ and $G$ be the $n$-dimensional algebraic torus on $\mathbf{k}$. We compute the weight spaces of $T_{\mathbf{A}/\mathbf{k}} (I)$ considered as a representation of $G$. Using this, we show that $T_{\mathbf{A}/\mathbf{k}} (I)$ preserves the integral closure of $I$ and the multiplier ideals of $I$.