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

Abstract

In this paper, we obtain the trace functions of Mantaci-Reutenauer algebra $\mathcal{MR}(W_{n})$, where $(W_{n},S_{n})$ is a Coxeter system of type $B_{n}$. We also show for every $\lambda \in \mathcal{DP}(n)$ that each characteristic class function $e_{\lambda}$ of the group $W_{n}$ is a trace function of Mantaci-Reutenauer algebra, where $\mathcal{DP}(n)$ stands for the set of all double partitions of $n$. Since the dimension of the trace function space on the Mantaci-Reutenauer algebra is $|\mathcal{DP}(n)|$, 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

References

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