论文标题
关于有限的阿贝尔P组的积极猜想
On the Positivity Conjecture for Finite Abelian p-Groups
论文作者
论文摘要
对于分区，$ \usewislineλ=（λ_{1}^{ρ_1}>λ_{2}^{ρ_2}>λ_{3}^}^{ρ_3}> \ ldots> \ ldots> \λ__{k} $ \ MATHCAL {a} _ {\luesdlineλ} = \ underSet {i = 1} {\ oftset {k} {\ oplus}}（\ nathcal {r}/π^{r}/π^{λ_i} {λ_i} \ mathcal {r}离散评估环，具有均匀的元素$π$生成的最大理想，具有有限的残基字段$ {\ bf k} = \ Mathcal {r}/π\ Mathcal {r} \ cong \ cong \ cong \ cong \ mathbb {f} _q $ \ Mathcal {g} _ {\luesdlineλ} \ backslash \ big（\ Mathcal {\ Mathcal {a} _ {\liseplineλ} \ times \ times \ times \ times \ times \ mathcal {a} _ {\useverlineλ} \ big big） $ \ MATHCAL {g} _ {\lundeslineλ} = aut（\ Mathcal {a} _ {\liseplineλ}）$，是带有整数系数的$ q $中的多项式。阳性猜想指出，这些系数实际上是非负的。在本文中，我们证明了这个猜想。
For a partition $\underlineλ = (λ_{1}^{ρ_1}>λ_{2}^{ρ_2}>λ_{3}^{ρ_3}>\ldots>λ_{k}^{ρ_k})$ and its associated finite $\mathcal{R}$-module $\mathcal{A}_{\underlineλ}=\underset{i=1}{\overset{k}{\oplus}} (\mathcal{R}/π^{λ_i}\mathcal{R})^{ρ_i}$, where $\mathcal{R}$ is a discrete valuation ring, with maximal ideal generated by a uniformizing element $π$, having finite residue field ${\bf k}=\mathcal{R}/π\mathcal{R}\cong \mathbb{F}_q$, the number of orbits of pairs $n_{\underlineλ}(q)= \mid \mathcal{G}_{\underlineλ}\backslash \big(\mathcal{A}_{\underlineλ}\times \mathcal{A}_{\underlineλ}\big)\mid$ for the diagonal action of the automorphism group $\mathcal{G}_{\underlineλ}= Aut(\mathcal{A}_{\underlineλ})$, is a polynomial in $q$ with integer coefficients. Positivity conjecture states that these coefficients are in fact non-negative. In this article, we prove this conjecture.
