Difference of Hilbert series of homogeneous monoid algebras and their normalizations

Let Q be an affine monoid, \(\Bbbk [Q]\) $\mathbb{k}[Q]$ the associated monoid \(\Bbbk \) $\mathbb{k}$ -algebra, and \(\Bbbk [\overline{Q}]\) $\mathbb{k}[\overline{Q}]$ its normalization, where we let \(\Bbbk \) $\mathbb{k}$ be a field. We discuss a difference of the Hilbert series of \(\Bbbk [Q]\) $\mathbb{k}[Q]$ and \(\Bbbk [\overline{Q}]\) $\mathbb{k}[\overline{Q}]$ in the case where \(\Bbbk [Q]\) $\mathbb{k}[Q]$ is homogeneous (i.e., standard graded). More precisely, we prove that if \(\Bbbk [Q]\) $\mathbb{k}[Q]$ satisfies Serre’s condition \((S_2)\) $(S_2)$ , then the degree of the h-polynomial of \(\Bbbk [Q]\) $\mathbb{k}[Q]$ is always greater than or equal to that of \(\Bbbk [\overline{Q}]\) $\mathbb{k}[\overline{Q}]$ . Moreover, we also show counterexamples of this statement if we drop the assumption \((S_2)\) $(S_2)$ .