Hook fusion procedure for direct product of symmetric groups

TL;DR

The paper addresses obtaining diagonal matrix elements of irreducible representations for the direct product $S_r \times S_s$ by extending Grime's hook fusion procedure. It introduces a fusion framework based on Jucys–Murphy elements and bi-partition indexing, and constructs a rational function $\Phi_{\boldsymbol{T}}(z)$ whose restriction to the hook subspace yields the diagonal element $F_{\boldsymbol{T}}$; the function factorizes into independent blocks corresponding to $S_r$ and $S_s$, leading to the exact expression $F_{\boldsymbol{T}} = (\sum_{g\in S_r} \langle v_{\boldsymbol{T}}, g v_{\boldsymbol{T}}\rangle g)(\sum_{g\in S_s} \langle v_{\boldsymbol{T}}, g v_{\boldsymbol{T}}\rangle g)$. This provides a parameter-efficient method to construct a complete set of orthogonal primitive idempotents for $C[S_r \times S_s]$ and extends the hook fusion technique beyond symmetric groups. The results enhance computational access to diagonal matrix elements and representation-theoretic structure in product groups, with potential applications in related algebras and fusion constructions.

Abstract

In this work, we derive a new expression for the diagonal matrix elements of irreducible representations of the direct product group $S_r\times S_s$ using Grime's hook fusion procedure for symmetric groups, which simplifies the fusion procedure by reducing the number of auxiliary parameters needed. By extending this approach to the product group setting, we provide a method for constructing a complete set of orthogonal primitive idempotents.

Figures (1)

• Figure 1: Bratteli diagram for $\mathbb{C}[S_3\times S_2]$

Theorems & Definitions (5)

• Example 1
• Proposition 2.1
• proof
• Theorem 3.1
• proof