When is a Specht ideal Cohen–Macaulay?
When is a Specht ideal Cohen–Macaulay?
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DOI：
10.1216/jca.2021.13.589
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发表时间：
2019-02
影响因子：
0.6
通讯作者：
Kohji Yanagawa
中科院分区：
文献类型：
--
作者：
Kohji Yanagawa
对于$n$的划分$\lambda$，设$i^{\rm Sp}_\lambda$是由形状为$\lambda$的所有Speht多项式生成的$R=K[x_1，\ldots，x_n]$的理想。我们证明了如果$R/i^{\rMSp}_\lambda$是Cohen-Macaulay，则$\lambda$的形式为$(a，1，ldots，1)$，$(a，b)$，或$(a，a，1)$。我们还证明了当${Rm char}(K)=0$时，反之亦然。来展示后一种说法，这些理想的激进性和Etingof等人的结果。是至关重要的。我们还指出，$R/i^{\Rm Sp}_{(n-3，3)}$不是Cohen-Macaulay当且仅当${\Rm char}(K)=2$。
For a partition $\lambda$ of $n$, let $I^{\rm Sp}_\lambda$ be the ideal of $R=K[x_1, \ldots, x_n]$ generated by all Specht polynomials of shape $\lambda$. We show that if $R/I^{\rm Sp}_\lambda$ is Cohen--Macaulay then $\lambda$ is of the form either $(a, 1, \ldots, 1)$, $(a,b)$, or $(a,a,1)$. We also prove that the converse is true if ${\rm char}(K)=0$. To show the latter statement, the radicalness of these ideals and a result of Etingof et al. are crucial. We also remark that $R/I^{\rm Sp}_{(n-3,3)}$ is NOT Cohen--Macaulay if and only if ${\rm char}(K)=2$.