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| Format: | Preprint |
| Publié: |
2026
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| Accès en ligne: | https://arxiv.org/abs/2605.05117 |
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| _version_ | 1866911652887134208 |
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| author | Wang, Xuan Zhang, Hanbin |
| author_facet | Wang, Xuan Zhang, Hanbin |
| contents | 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 $\text{imm}_λ(\mathcal M_G)$ be the immanant of $\mathcal M_G$ with respect to a partition $λ$ and $\mathcal I_λ(G)$ be the number of formally different monomials occurring in $\text{imm}_λ(\mathcal M_G)$ (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 $$ \text{imm}_{(4,1^{n-4})}(\mathcal M_G)=\text{imm}_{(2,2,2,1^{n-6})}(\mathcal M_G).$$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_05117 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On immanants of Cayley tables Wang, Xuan Zhang, Hanbin Combinatorics 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 $\text{imm}_λ(\mathcal M_G)$ be the immanant of $\mathcal M_G$ with respect to a partition $λ$ and $\mathcal I_λ(G)$ be the number of formally different monomials occurring in $\text{imm}_λ(\mathcal M_G)$ (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 $$ \text{imm}_{(4,1^{n-4})}(\mathcal M_G)=\text{imm}_{(2,2,2,1^{n-6})}(\mathcal M_G).$$ |
| title | On immanants of Cayley tables |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.05117 |