Generalized Burnside Algebra of type $B_{n}$

TL;DR

This paper introduces a generalized Burnside algebra for type B Coxeter groups, constructs algebra morphisms, and provides methods to compute conjugacy class sizes and element counts efficiently.

Contribution

It defines a new algebraic structure for type B Coxeter groups and develops tools for analyzing conjugacy classes and element enumeration.

Findings

Constructed a surjective algebra morphism between Mantaci-Reutenauer and Burnside algebras.

Derived formulas for counting conjugacy classes of type B groups.

Developed an effective method to determine the size of elements of type S_n.

Abstract

We define the generalized Burnside algebra $HB(W_{n})$ for $B_{n}$-type Coxeter group $W_{n}$ and construct an surjective algebra morphism between Mantaci-Reutenauer algebra ${\sum}'(W_{n})$ and $HB(W_{n})$. Then, by obtaining the primitive idempotents $(e_{\lambda})_{\lambda \in \mathcal{DP}(n)}$ of $HB(W_{n})$, we consider the image $\textrm{res}_{W_{A}}^{W_{n}}e_{B}$ and $\textrm{ind}_{W_{A}}^{W_{n}}e_{B}^{A}$ under restriction and induction map between generalized Burnside algebras. We give an alternative formula to compute the elements number of conjugate classes of $W_{n}$. We also obtain an effective method to determine the size of $\mathcal{C}(S_{n})$ which is the set of elements of type $S_{n}$.