The fundamental group $\pi_{1}(\operatorname{star}_{\Sigma}(\hat{\sigma},\hat{\tau})\backslash D)$ is a free group with $k\cdot l$ generators. In a suitable basis the monodromy

$$T\colon\pi_{1}(\operatorname{star}_{\Sigma}(\hat{\sigma},\hat{\tau})\backslash D)\rightarrow\operatorname{SL}(d-1,\mathbb{Z})$$

can be represented (faithfully on the abelianization of $\pi_{1}$) by an index $\operatorname{vol}(\sigma)\cdot\operatorname{vol}(\tau)$ subgroup of the (abelian) group of matrices in the form

$$\left(\begin{array}[]{cccccc}1&0&\ldots&*&\ldots&*\\ 0&1&\ldots&\vdots&\ddots&\vdots\\ 0&0&\ddots&*&\ldots&*\\ \vdots&\vdots&\ddots&\ddots&\vdots&\vdots\\ 0&0&\ldots&0&1&0\\ 0&0&\ldots&0&0&1\\ \end{array}\right)$$

(the identity matrix plus an $l\times k$ block in the right upper corner).