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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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