On immanants of Cayley tables

Abstract

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 ${\text{imm}}_{\hat{I}}»({\mathcal{M}}_G)$ be the immanant of ${\mathcal{M}}_G$ with respect to a partition $\hat{I}»$ and ${\mathcal{I}}_{\hat{I}}»(G)$ be the number of formally different monomials occurring in ${\text{imm}}_{\hat{I}}»({\mathcal{M}}_G)$ (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 $|G|\equiv 2(\mathrm{mod}4)$, then $${\mathcal{I}}_{(n-1,1)}(G)=\mathcal{P}(G)\text{and}{\mathcal{I}}_{(2,1^{n-2})}(G)=\mathcal{D}(G).$$ (3) If $|G|$ is odd and $|G|\ge 7$, then $${\text{imm}}_{(4,1^{n-4})}({\mathcal{M}}_G)={\text{imm}}_{(2,2,2,1^{n-6})}({\mathcal{M}}_G).$$