Difference of Hilbert series of homogeneous monoid algebras and their normalizations

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