Wreath product action on generalized Boolean algebras

TL;DR

This paper studies the action of wreath products on generalized Boolean algebras and provides an explicit block diagonalization of the commutant, advancing understanding of symmetry and representation in algebraic combinatorics.

Contribution

It introduces an explicit block diagonalization of the commutant for wreath product actions on generalized Boolean algebras, extending previous algebraic combinatorics results.

Findings

Explicit block diagonalization of the commutant

Analysis of wreath product actions on Boolean algebras

Enhanced understanding of symmetry in algebraic combinatorics

Abstract

Let G be a finite group acting on the finite set X such that the corresponding (complex) permutation representation is multiplicity free. There is a natural rank and order preserving action of the wreath product G~S_n on the generalized Boolean algebra B_X(n). We explicitly block diagonalize the commutant of this action.