A canonical polytopal resolution for transversal monomial ideals

Let $S = k[x_{11}, \cdots, x_{1b_1}, \cdots, x_{n1}, \cdots, x_{nb_n}]$ be a polynomial ring in $m = b_1 + \cdots + b_n$ variables over a field $k$. For all $j$, $1\le j \le n$, let $P_j$ be the prime ideal generated by variables $\{x_{j1}, \cdots, x_{jb_j}\}$ and let $$I_{n, t} = \sum_{1\le j_1< \cdots <j_t\le n} P_{j_1}\ldots P_{j_t}$$ be the transversal monomial ideal of degree $t$ on $P_1, \cdots, P_n$. We explicitly construct a canonical polytopal $\mathbb{Z}^t$-graded minimal free resolution for the ideal $I_{n, t}$ by means of suitable gluing of polytopes.