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Partition algebras \(\mathsf {P}_k(n)\) with \(2k>n\) and the fundamental theorems of invariant theory for the symmetric group \(\mathsf {S}_n\). (English) Zbl 1411.05274
Authors’ abstract: Assume \(\mathsf {M}_n\) is the \(n\)-dimensional permutation module for the symmetric group \(\mathsf {S}_n\), and let \(\mathsf {M}_n ^{\otimes k}\) be its \(k\)-fold tensor power. The partition algebra \(\mathsf {P}_k(n)\) maps surjectively onto the centralizer algebra \(\mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\) for all \(k,n\in \mathbb Z_{\geqslant 1}\) and isomorphically when \(n\geqslant 2k\). We describe the image of the surjection \(\Phi_{k,n}:\mathsf {P}_k(n)\rightarrow \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes}k)\) explicitly in terms of the orbit basis of \(\mathsf {P}_k(n)\) and show that when \(2k>n\) the kernel of \(\Phi_{k,n}\) is generated by a single essential idempotent \(\mathsf e_{k,n}\), which is an orbit basis element. We obtain a presentation for \(\mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\) by imposing one additional relation, \(\mathsf e_{k,n}=0\), to a presentation of the partition algebra \(\mathsf {P}_k(n)\) when \(2k>n\). As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group \(\mathsf {S}_n\). We show under the natural embedding of the partition algebra \(\mathsf {P}_n(n)\) into \(\mathsf {P}_k(n)\) for \(k \geq n\) that the essential idempotent \(\mathsf e_{n,n}\) generates the kernel of \(\Phi_{k,n}\). Therefore, the relation \(\mathsf e_{n,n}=0\) can replace \(\mathsf e_{k,n}=0\) when \(k\geqslant n\).

MSC:
05E10 Combinatorial aspects of representation theory
20C30 Representations of finite symmetric groups
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