Hook fusion procedure for hyper-octahedral groups

TL;DR

This work develops a hook fusion procedure for the hyperoctahedral group $\mathbb{Z}_{2}\wr S_{n}$ (Coxeter type $B$) by adapting Grime’s hook fusion method to minimize auxiliary parameters in the fusion process. It constructs Baxterized and Jucys–Murphy–type elements to form a rational fusion function $\Phi_{\bm{T}}$ whose restriction to the principal hook subspace $H_{\bm{\lambda}}$ is regular, and its evaluation yields the diagonal matrix elements $F_{\bm{T}}$ up to a scalar factor $1/2^{n}$, delivering a complete set of pairwise orthogonal primitive idempotents. The approach reduces the number of auxiliary parameters relative to prior fusion procedures and provides explicit factorization properties (left-divisibility by $s_i\pm1$) tied to row/column adjacencies, enhancing practical computation of irreducible representations for the type-$B$ Coxeter group. These results unify Yang–Baxter fusion methods with the representation theory of hyperoctahedral groups, offering a streamlined route to diagonal elements and idempotents useful in applications spanning algebraic combinatorics and mathematical physics.

Abstract

We derive a new expression for the diagonal matrix elements of irreducible representations of the hyperoctahedral group. This expression is obtained using Grime's hook fusion procedure for symmetric groups, which minimizes the number of auxiliary parameters required in the fusion process.

Theorems & Definitions (17)

• Definition 2.1
• Theorem 2.2
• Definition 3.1
• Example 1
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Proposition 3.4
• proof