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Partition Algebras and the Invariant Theory of the Symmetric Group

  • Georgia BenkartEmail author
  • Tom Halverson
Chapter
Part of the Association for Women in Mathematics Series book series (AWMS, volume 16)

Abstract

The symmetric group \(\mathsf {S}_n\) and the partition algebra \(\mathsf {P}_k(n)\) centralize one another in their actions on the k-fold tensor power \(\mathsf {M}_n^{\otimes k}\) of the n-dimensional permutation module \(\mathsf {M}_n\) of \(\mathsf {S}_n\). The duality afforded by the commuting actions determines an algebra homomorphism \(\varPhi _{k,n}: \mathsf {P}_k(n) \rightarrow \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\) from the partition algebra to the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), which is a surjection for all  \(k, n \in \mathbb {Z}_{\ge 1}\), and an isomorphism when \(n \ge 2k\).   We present results that can be derived from the duality between \(\mathsf {S}_n\) and \(\mathsf {P}_k(n)\), for example, (i) expressions for the multiplicities of the irreducible \(\mathsf {S}_n\)-summands of \(\mathsf {M}_n^{\otimes k}\), (ii) formulas for the dimensions of the irreducible modules for the centralizer algebra \( \mathsf {End}_{\mathsf {S}_n}(\mathsf {M}_n^{\otimes k})\), (iii) a bijection between vacillating tableaux and set-partition tableaux, (iv) identities relating Stirling numbers of the second kind and the number of fixed points of permutations, and (v) character values for the partition algebra \(\mathsf {P}_k(n)\). When \(2k >n\), the map \(\varPhi _{k,n}\) has a nontrivial kernel which is generated as a two-sided ideal by a single idempotent. We describe the kernel and image of \(\varPhi _{k,n}\) in terms of the orbit basis of \(\mathsf {P}_k(n)\) and explain how the surjection \(\varPhi _{k,n}\) can also be used to obtain the fundamental theorems of invariant theory for the symmetric group.

Keywords

Symmetric group Partition algebra Schur–Weyl duality Invariant theory 

Mathematics Subject Classification (2010)

MSC 05E10 MSC 20C30 

Notes

Acknowledgements

The authors thank the referee for a careful proofreading and useful suggestions.

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Copyright information

© The Author(s) and the Association for Women in Mathematics 2019

Authors and Affiliations

  1. 1.University of Wisconsin - MadisonMadisonUSA
  2. 2.Macalester CollegeSaint PaulUSA

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