Alexander duality for the alternative polarizations of strongly stable ideals

We will define the Alexander duality for strongly stable ideals. More precisely, for a strongly stable ideal $I \subset \Bbbk[x_1, \ldots, x_n]$ with ${\rm deg}(\mathsf{m}) \le d$ for all $\mathsf{m} \in G(I)$, its dual $I^* \subset \Bbbk[y_1, \ldots, y_d]$ is a strongly stable ideal with ${\rm deg}(\mathsf{m}) \le n$ for all $\mathsf{m} \in G(I^*)$. This duality has been constructed by Fl$\o$ystad et al. in a different manner, so we emphasis applications here. For example, we will describe the Hilbert serieses of the local cohomologies $H_\mathfrak{m}^i(S/I)$ using the irreducible decomposition of $I$ (through the Betti numbers of $I^*$).