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Zero-dimensional Gorenstein algebras with the action of the symmetric group. (English) Zbl 1173.13005
For $$K$$ an algebraically closed field of characteristic $$0$$, let $$A=A\left( n,k\right) =K\left[ x_{1},\dots,x_{k}\right] /\left( x_{1}^{n},\dots ,x_{k}^{n}\right) .$$ Then $$A\cong\left( K^{n}\right) ^{\otimes k}$$ as $$\text{GL}\left( n\right) \times S_{k}$$-modules (here $$S_{k}$$ is the symmetric group, acting via permutation of variables). The element $$l:=x_{1}+\cdots+x_{k}\in A$$ is a strong Lefschetz element. The ring of invariants under $$G:=S_{k}$$ is shown to be $$A^G=K[e_1,\dots,e_k] /(p_n,p_{n+1},\dots,p_{n+k-1})$$, where $$e_{d}$$ is the elementary symmetric polynomial of degree $$d$$ and $$p_{d}=x_{1}^{d}+\cdots+x_{k}^{d}.\;$$The Hilbert series of $$A^{G}$$ is quickly derived, and this series shows that $$\dim A^{G}=\binom{n+k-1}{k}.$$
Let $$\lambda=\left( k_{1},\dots,k_{r}\right) \vdash k,$$ $$r\leq n$$ be a partition of $$k,$$ and let $$Y^{\lambda}\left( A\right)$$ be the Young summetrizer corresponding to $$\lambda.$$Then the Hilbert series of $$Y^{\lambda }\left( A\right)$$ is described as a graded subspace of $$A.$$ Let $$W^{\lambda}$$ be an irreducible $$\text{GL}\left( n\right)$$-module corresponding to $$\lambda.$$ Let $$\phi^{\lambda}:\text{GL}\left( n\right) \rightarrow\text *{GL}\left( W^{\lambda}\right)$$ be the corresponding irreducible representation of $$\text{GL}\left( n\right) .$$ For any $$a\in K$$ let $$J\left( a,n\right)$$ be the usual $$n\times n$$ Jordan block. Then the Jordan canonical form of $$\phi^{\lambda }\left( J\left( a,n\right) \right)$$ is given, described in terms of the dual Hilbert series.

MSC:
 13A50 Actions of groups on commutative rings; invariant theory
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References:
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