Artinian Gorenstein rings with the action of the symmetric group

我们让对称群作用在多项式环上，并利用Weyl对偶得到了对称群的Young子群的不变余代数的某些结果。作为结果，我们获得了以下结果。(1)引入了高阶Hessians的概念，并利用它刻画了Artin-Gorenstein局部环中的强Lefschetz元。(2)证明了由Young子群构成的不变量余代数具有强Lefschetz性质。(3)得到了多项式环上微分模的外代数的不可约分解。(4)给出了三元幂对称函数的一些性质，并对幂和形成正则序列的次数选择问题进行了猜想。

We let the symmetric group act on the polynomial ring, and used the Weyl duality to obtain certain results on the invariant coalgebras by Young subgroups of the symmetric group. As results, we obtained the following. (1) We introduced the notion of higher Hessians and using it, we characterized the strong Lefschetz elements in Artinian Gorenstein local rings. (2) We proved that the coalgebras of invariants by the Young subgroups have the strong Lefschetz property. (3) We obtained the irreducible decomposition of the exterior algebra of the differential module over the polynomial ring. (4) We showed some properties of power some symmetric functions in three variables and made a conjecture to the problem to ask in which choice of degrees the power sums form a regular sequence.