A Note About the Trace Functions on Mantaci--Reutenauer Algebra

Authors : Hasan ARSLAN

Pages : 140-143

View : 15 | Download : 12

Publication Date : 2018-04-15

Article Type : Research Paper

Abstract :In this paper, we obtain the trace functions of Mantaci-Reutenauer algebra $\mathcal{MR}insert ignore into journalissuearticles values(W_{n});$, where $insert ignore into journalissuearticles values(W_{n},S_{n});$ is a Coxeter system of type $B_{n}$. We also show for every $\lambda \in \mathcal{DP}insert ignore into journalissuearticles values(n);$ that each characteristic class function $e_{\lambda}$ of the group $W_{n}$ is a trace function of Mantaci-Reutenauer algebra, where $\mathcal{DP}insert ignore into journalissuearticles values(n);$ stands for the set of all double partitions of $n$. Since the dimension of the trace function space on the Mantaci-Reutenauer algebra is $|\mathcal{DP}insert ignore into journalissuearticles values(n);|$, it exactly coincides with the algebra $\mathbb{Q}\textrm{Irr}W_{n}$ generated by the irreducible characters of the group $W_{n}$. Although the multiplication of basis elements $d_{A}$ and $d_{A`}$ is not commutative in Mantaci-Reutenauer algebra, the images of $d_{A}d_{A`}$ and $d_{A`}d_{A}$ under $e_{\lambda}$ are equal to each other.
Keywords : Mantaci Reutenauer algebra, Orthogonal primitive idempotent, Trace function