Projection formulas and a refinement of Schur--Weyl--Jones duality for symmetric groups
Ewan Cassidy
TL;DR
The paper advances Schur–Weyl–Jones theory by constructing a refined subspace A_k(n) of (C^n)^{⊗k} that carries a commuting S_n × S_k action and, for n ≥ 2k, admits a clean decomposition A_k(n) ≅ ⊕_{λ ⊢ k} V^{λ^{+}(n)} ⊗ V^{λ}. It then isolates explicit irreducible blocks U_{λ^{+}(n)} through the vector ξ_{λ}^{norm} and proves that the S_{2k} × S_k orbit of ξ_{λ}^{norm} generates a block isomorphic to V^{λ^{+}(2k)} ⊗ V^{λ} with multiplicity one, enabling an explicit orthogonal projection Q_{λ,n} onto U_{λ^{+}(n)}. The projection is computed via Weingarten calculus and expressed as a Weingarten-type sum over partitions and permutations, providing a practical formula to extract irreducible characters χ^{λ^{+}(n)} by traces against Q_{λ,n}. The work yields a recursive dimension formula for A_k(n) and suggests applications to the study of word maps and expected characters of w-random permutations, bridging combinatorial representation theory with probabilistic questions in symmetric groups.
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 $\left(\mathbb{C}^n\right)^{\otimes k}.$ We give a refinement of this, determining a subspace of $\left(\mathbb{C}^n\right)^{\otimes k}$ 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 $\left(\mathbb{C}^n\right)^{\otimes k}$ that are isomorphic to certain irreducible representations of $S_{n}\times S_{k}.$ We then use the Weingarten calculus for the symmetric group to obtain an explicit formula for the orthogonal projection from $\left(\mathbb{C}^n\right)^{\otimes k}$ to each subspace.