On immanants of Cayley tables

Let $G$ be a finite abelian group of order $n$ and let $\mathcal M_G=(x_{a+b})_{a,b\in G}$ be the Cayley table of $G$. Let $\text{imm}_\lambda(\mathcal M_G)$ be the immanant of $\mathcal M_G$ with respect to a partition $\lambda$ and $\mathcal I_\lambda(G)$ be the number of formally different monomials occurring in $\text{imm}_\lambda(\mathcal M_G)$ (in particular, we denote by $\mathcal P(G)$ (resp. $\mathcal D(G)$) for the corresponding quantity for $\text{per}(\mathcal M_G)$ (resp. $\text{det}(\mathcal M_G)$) for simplicity). The study of $\mathcal P(G)$ and $\mathcal D(G)$ lies at the intersection of algebraic combinatorics and additive combinatorics. In this paper, we prove the following results. (1) If $|G|$ is a prime power, then $$\mathcal P(G)=\mathcal D(G).$$ (2) If $|G|$ is odd, then $$\mathcal I_{(n-1,1)}(G)= \mathcal I_{(2,1^{n-2})}(G)=0,$$ and if $|G|\equiv 2\pmod 4$, then $$\mathcal I_{(n-1,1)}(G)=\mathcal P(G)\quad \text{and}\quad \mathcal I_{(2,1^{n-2})}(G)=\mathcal D(G).$$ (3) If $|G|$ is odd and $|G|\ge 7$, then $$\text{imm}_{(4,1^{n-4})}(\mathcal M_G)=\text{imm}_{(2,2,2,1^{n-6})}(\mathcal M_G).$$