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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.

13A50 Actions of groups on commutative rings; invariant theory
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