Abstract
In this paper, we have first presented the construction of the linear characters of a finite Coxeter group $G_n$ of type $B_n$ by lifting all linear characters of the quotient group $G_n/[G_n,G_n]$ of the commutator subgroup $[G_n,G_n]$. Also we show that the sets of distinguished coset representatives $D_A$ and $D_{A^{\prime}}$ for any two signed compositions A, A' of n which are $G_n$-conjugate to each other and for each conjugate class ${\mathcal{C}}_{\lambda}$ of $G_n$, where ${\lambda}{\in}\mathcal{BP}(n)$, the equality ${\mid}{\mathcal{C}}_{\lambda}{\cap}D_A{\mid}={\mid}{\mathcal{C}}_{\lambda}{\cap}D_{A^{\prime}}{\mid}$ holds. Finally, we have given the general structure of units of Mantaci-Reutenauer algebra.