Let G be a finite abelian group of order n. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\mathcal {M}}_G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> be the Cayley table of G and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textsf{per}({\mathcal {M}}_G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">per</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the permanent of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {M}}_G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation>. An interesting result of Hall provided a one to one correspondence between the monomials in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\textsf{per}({\mathcal {M}}_G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="sans-serif">per</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and zero-sum sequences over G of length n. Generalizing Hall’s result, Panyushev conjectured an analogous correspondence concerning the generalized Cayley table of G. In this paper, we disprove Panyushev’s conjecture and provide a general characterization of the aforementioned correspondence. As the permanent and determinant matrix functions are special cases of immanants (which are very important objects in algebraic combinatorics), we also provide some discussions on the immanants of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathcal {M}}_G$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>G</mi> </msub> </math></EquationSource> </InlineEquation> and propose some interesting conjectures.

On Immanants of the Cayley Table of Finite Abelian Groups

期刊论文
发表日期：

2026年1月20日

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Xuan Wang,
Hanbin Zhang,
Shiwen Zhang

摘要

Let G be a finite abelian group of order n. Let ${\mathcal {M}}_G$ $M_{G}$ be the Cayley table of G and $\textsf{per}({\mathcal {M}}_G)$ $per (M_{G})$ the permanent of ${\mathcal {M}}_G$ $M_{G}$ . An interesting result of Hall provided a one to one correspondence between the monomials in $\textsf{per}({\mathcal {M}}_G)$ $per (M_{G})$ and zero-sum sequences over G of length n. Generalizing Hall’s result, Panyushev conjectured an analogous correspondence concerning the generalized Cayley table of G. In this paper, we disprove Panyushev’s conjecture and provide a general characterization of the aforementioned correspondence. As the permanent and determinant matrix functions are special cases of immanants (which are very important objects in algebraic combinatorics), we also provide some discussions on the immanants of ${\mathcal {M}}_G$ $M_{G}$ and propose some interesting conjectures.