On the determinant of multiplication map of a monomial complete intersection ring

In this article, we consider the monomial complete intersection algebra $\mathbb{K}[x,y]/\langle x^d,y^q\rangle$ in two variables. For elements $l_1,\ldots,l_{d+q-2k}$ of degree $1$, we give a formula of the deteminant of linear map from the homogeneous component of degree $k$ to the homogenous component of degree $d+q-k$ defined by the multiplication of $l_1 \cdots l_{d+q-2k}$.