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 $I^*$).