Projection formulas and a refinement of Schur--Weyl--Jones duality for symmetric groups

2023/12/04 by Ewan Cassidy, Cassidy, Ewan · 1 citation
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2312.01839

openalex publication_date 2023/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

Schur--Weyl--Jones duality establishes the connection between the commuting actions of the symmetric group S_n and the partition algebra P_k(n) on the tensor space (ℂⁿ)^{⊗ k}. We give a refinement of this, determining a subspace of (ℂⁿ)^{⊗ k} on which we have a version of Schur--Weyl duality for the symmetric groups S_n and S_k. We use this refinement to construct subspaces of (ℂⁿ)^{⊗ k} that are isomorphic to certain irreducible representations of S_n× S_k. We then use the Weingarten calculus for the symmetric group to obtain an explicit formula for the orthogonal projection from (ℂⁿ)^{⊗ k} to each subspace.

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